THE THEORY OF RELATIVITY

MACMILLAN AND CO., LIMITED

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TORONTO

THE THEORY OF
RELATIVITY

BY

L. SILBERSTEIN, PH.D.

LECTURER IN NATURAL PHILOSOPHY AT THE UNIVERSITY OF ROME

MACMILLAN AND CO., LIMITED

ST. MARTIN'S STREET, LONDON

1914

COPYRIGHT

PREFACE

THIS introduction to the Theory of Relativity is based in part
upon a course of lectures delivered in University College, London,
1912-13. The treatment, however, has been made much more
systematical, and the subject matter has been extended very con-
siderably; but, throughout, the attempt has been made to confine
the reader's attention to matters of prime importance. With this
aim in view, many particular problems even of great interest have
not been touched upon. On the other hand, it seemed advantageous
to trace the connexion of the modern theory with the theories
and ideas that preceded it. And the first three chapters, therefore,
are devoted to the fundamental ideas of space and time underlying
classical physics, and to the electromagnetic theories of Maxwell,
Hertz-Heaviside and Lorentz, from the last of which Einstein's
theory of relativity was directly derived. In the exposition of the
theory itself free use has been made not only of the matrix method
of representation employed by Minkowski, but even more of the
language of quaternions. Very little indeed of these mathematical
methods is required to follow the exposition, and this little is
given in Chapter V., in a form which will be at once accessible
to those acquainted with the elements of the ordinary vector
algebra.

It is hoped that the book will give the reader a good insight
into the spirit of the theory and will enable him easily to follow
the more subtle and extended developments to be found in a
large number of special papers by various investigators.

vj PREFACE

I gladly take the opportunity of expressing my thanks to Mr.
William Francis and Dr. T. Percy Nunn for their kindness in
reading a large portion of the MS., to Prof. Alfred W. Porter, F.R.S.,
for reading all the proofs and for many valuable suggestions,
and to the Publishers and the Printers for the care they have
bestowed on my work.

L. S.

LONDON, April, 1914.

CLASSICAL RELATIVITY -

CONTENTS

CHAPTER I

PAGE

i

CHAPTER II

p.*+

MAXWELLIAX EQUATIONS FOR MOVING MEDIA AND FRESNEL'S

DRAGGING COEFFICIENT. LORENTZ'S EQUATIONS 21

CHAPTER III

THEOREM OF CORRESPONDING STATES. SECOND-ORDER DIFFICULTIES.
THE CONTRACTION HYPOTHESIS. LORENTZ'S GENERALIZED
THEORY 64

CHAPTER IV

EINSTEIN'S DEFINITION OF SIMULTANEITY. THE PRINCIPLES OF
RELATIVITY AND OF CONSTANT LIGHT- VELOCITY. THE
LORENTZ TRANSFORMATION 107 -142- 92

CHAPTER V

VARIOUS REPRESENTATIONS OF THE LORENTZ TRANSFORMATION 123

CHAPTER VI

COMPOSITION OK VELOCITIES AND THE LORENTZ GROUP 163

CHAPTER VII

PHYSICAL QUATERNIONS. DYNAMICS OF A PARTICLE 182

viii CONTENTS

CHAPTER VIII

FUNDAMENTAL ELECTROMAGNETIC EQUATIONS 205

CHAPTER IX

ELECTROMAGNETIC STRESS, ENERGY AND MOMENTUM. EXTENSION

TO GENERAL DYNAMICS 232

CHAPTER X

MINKOWSKIAN ELECTROMAGNETIC EQUATIONS FOR PONDERABLE

MEDIA 260

INDEX - ----- 290

CHAPTER I.

CLASSICAL RELATIVITY.

BEFORE entering upon the subject proper of this volume, namely, the
modern doctrine of Relativity and the history of its origin and develop-
ment, it seems desirable to dwell a little on the more familiar ground
of what might be called the classical relativity, and to consider

CORRIGENDA

60, line 4 from the foot, for on read ou

65, line 8, for P read P'

71, „ ^for */<* read tf/c*
119, ,, 14, for 0(z/) = i read 0(z;)=i/a
149, line 9 from the foot, for W^ W.2 read
153. » 7 „ „ for A2 read Az
189, line 3, for right-hand read left-hand
222, line 2 from the foot, for vectors read vector operators
229, line 13, interchange the words 'real' and 'imaginary '
235, „ 20, for P'p' read (P'p')
235> „ 25,>r/n/ readl'n,
276, ,, 6, for i/c read i/c

viii CONTENTS

CHAPTER VIII

FUNDAMENTAL ELECTROMAGNETIC EQUATIONS 205

CHAPTER IX

ELECTROMAGNETIC STRESS, ENERGY AND MOMENTUM. EXTENSION

TO GENERAL DYNAMICS - 232

CHAPTER X

MINKOWSKIAN ELECTROMAGNETIC EQUATIONS FOR PONDERABLE

MEDIA 260

INDEX ------ ----- 290

CHAPTER I.

CLASSICAL RELATIVITY.

BEFORE entering upon the subject proper of this volume, namely, the
modern doctrine of Relativity and the history of its origin and develop-
ment, it seems desirable to dwell a little on the more familiar ground
of what might be called the classical relativity, and to consider
two particular points which are of fundamental importance, not only
for the appreciation of the whole subject to follow, but also for an
adequate understanding of almost all physico-mathematical considera-
tions. What I am alluding to are the following questions: i° the
choice of a framework of axes or, more generally, of a system
of reference in space, and 2° the definition of physical time, or the
selection of a clock or time-keeper, to be employed for the quantita-
tive determination of a succession of physical events.

Both of these questions existed and were solved, at least implicitly,
a long time before the invention of the modern Principle of Relativity,
in fact centuries ago, in their essence as early as Copernicus founded
his system.*

The question of a space-framework is obvious enough and widely
known ; it will require therefore only a few simple remarks.

The most superficial observation of everyday life would suffice to
show that the form and the degree of simplicity of the statement of
the laws of physical phenomena, more especially of the laws of motion
of what are called material bodies, depend essentially on our selec-
tion of a system of reference in space. Certain frameworks of
reference are peculiarly fitted for the representation of a particular

*A clear and beautiful statement of the fundamental importance of the
Copernican idea is to be found in P. Painleve's article ' Mecanique ' in the collec-
tive volume De la Mtthode dans les Sciences, edited by Emile Borel. (Paris,
F. Alcan, 1910.)

S.R. A

£ {THEORY OF -RELATIVITY

instance of motion of a particular body or also of almost any
observable motion of bodies in general, leading to a high degree
of completeness, exactness and simplicity, while other frame-
works (moving in an arbitrary manner relatively to those) give of
the same phenomena a most complicated, intricate and confused
picture.*

Suppose that somebody, ignorant of the work of Copernicus,
Galileo and Newton, but otherwise gifted with the highest experi-
mental abilities and mathematical skill (a quite imaginary supposition,
being hardly consistent with the first one), chooses the interior of an
old-fashioned coach, driven along a fairly rough road, as his laboratory
and tries to investigate the laws of motion of bodies enclosed together
with him in the coach — say, of a pendulum or of a spinning top — and
selects that vehicle as his system of reference. Then his tangible
bodies and his conceptual * material points,' starting from rest or any
given velocity, would describe the most wonderful paths, in incessant
shocks and jerky motions; the axis of his 'free gyroscope' would oscillate
in a most complicated way, — never disclosing to him the constancy of
the vector known to us as the 'angular momentum,' i.e. the rotatory
analogue of Newton's first law of motion. Nor would the uniform
translational motion have for him any peculiarly simple or generally
noteworthy properties at all. His mechanical experience being, in a
word, full of surprises, he would soon give up his task of stating any
laws of motion whatever with reference to the coach. Getting out of
it on to firm ground, he will readily find out that the earth is a much
better system of reference. With this framework, smoothness and
simplicity will take the place of hopeless irregularity. Undoubtedly,
this property must have been remarked in a very early stage of man's
history, and the above example will appear to the least trained student
of mechanics of our present times trivial and simply ridiculous. ' Of
course,' he would say, ' the motions of material bodies relatively to
that coach are so very complicated, for that vehicle is itself moving in
a highly complicated way.' He would hardly consider it worth while
to add ' relatively to the earth.' The coach being such a small, insigni-
ficant thing in comparison with the terrestrial globe, it would seem
extravagant to our interlocutor, if somebody insisted rather on saying
that it is the earth which moves in such a complicated way relatively

* And as to 'absolute motion,' regardless of any system of reference, it is need-
less to mention that it is devoid of meaning in exactly the same way as ' absolute
position.'

FRAME OF REFERENCE 3

to the coach on its particular journey. But, as a matter of fact, both
of these reference-systems move relatively to one another, and the
comparative insignificance of one of them would, by itself, be but a
very feeble argument (as we shall see presently, from another
example).

At any rate the earth, the 'firm ground,' allowance being made
for occasional large shocks and for very small but incessant oscil-
lations of every part of its surface,* has proved to be an excellent
system of reference for almost all motions, especially those on a
small scale with regard to space and time, and practically without
any reservation for all pieces of machinery and technical contriv-
ance. In fact, the earth as a system of reference offered at once the
advantage of a high degree of simplicity of description of states of
equilibrium and motion, opening a wide field for the application of
Newton's mechanics, at least as regards purely terrestrial observations
and experiments.! The earth is then a reference-system which is
constantly used also by the most advanced modern student of
mechanics.

But things become altogether different when we look up to the sky
and desire to bring into our mechanical scheme also the motions of
those luminous points, the celestial bodies, including, of course, our
satellite, the moon, and our sun. Then the earth loses its privilege
as a framework of reference. If it were only for the so-called ' fixed
stars,' which form the enormous majority of those luminous points
(and the moon too), we could still satisfy our vanity and continue to
consider our globe as an universal mechanical system of reference,
the system of reference, as it were. On our plane drawings, or in our
three-dimensional models, we could then represent the earth by a
fixed disc, or sphere, respectively, with a smaller sphere moving
round it in a circular orbit, to imitate our moon, the whole sur-
rounded by a large spherical shell of glass sown with millions of tiny
stars, spinning gently and uniformly round the earth's axis, — very

* Which gave so much trouble to the late Sir G. H. Darwin and his brother in
their attempts to measure directly the gravitational action of the moon, as described
in Sir G. H. Darwin's attractive popular book, The Tides and Kindred Phenomena
in the Solar System, London, 1898 (German edition by A. Pockels, enlarged ;
Teubner, Leipzig, 1911).

t With the exception of those of the type of Foucault's pendulum experiments,
performed with the special purpose ' of showing the earth's rotation. ' In more
recent times the pendulum could be successfully replaced by a gyroscope, as
originally suggested, and tried, by Foucault himself.

4 THE THEORY OF RELATIVITY

much like, in fact, some primitive mental pictures of the universe.*
But the case becomes entirely different when we come to consider the
far less numerous class of luminous points or little discs, the planets,
and the comets, moving visibly among the ' fixed ' shining points
in a complicated way. Then, even before touching any dynamical
part of the celestial problem, we are compelled to give up our earth
as a system of reference and replace it by that of the ' fixed stars,'
originally so inconspicuous, or — what turns out to be equally good —
by a framework of axes pointing from an initial point fixed in the sun
towards any given triad of fixed stars. It is needless to tell here
again the long story of that admirable and ingenious system which
was founded by Ptolemy (born about 140 B.C.), which held the
field during fourteen centuries, to be replaced finally and definitely
by the system of Copernicus (1473-1543), which transferred to the
sun the previous dignity of the earth, f The Copernican system of
reference had the enormous advantage of simplicity, quite inde-
pendently of any mechanical, i.e. (to put it more strictly) dynamical
considerations. Its superiority to the geocentric system manifested
itself already in the simplicity it gave to the paths of the solar family
of bodies, the wonderfully simple shapes of the orbits of the planets.
In the geocentric scheme we had the complicated system of
'excentrics and epicycles' of Ptolemy, whereas taking, in our
drawing or model, the sun as fixed, the orbits of the planets became
simple circles, which in the next step of approximation turned out to
be slightly elliptic. Thus the Copernican system of reference had its
enormous advantages before any properly mechanical point of the
subject was entered upon. Historically, in fact, the mechanics of
Galileo and Newton came a long time after Copernicus, so that the

*The earth as the centre of the universe, with the ' crystal spheres,' with the
stars stuck to them, spinning round the earth, still formed part of the teachings of
the Ionian school of philosophers founded by Thales (born about 640 B.C.). The
first to suggest the rotation of the earth round its axis and its motion round the
sun seems to have been Pythagoras, one of Thales' disciples, though it has been
later unjustly attributed to Philolaus, one of Pythagoras' disciples (born about
450 B.C.).

t Although I do not claim to give here anything like a history of astronomy, it
may be worth mentioning that the Pythagoreans already taught that the planets
and comets were circling round the sun. But at any rate the Ptolemaean geo-
centric system reigned universally from the second till the fifteenth century, the
only serious objection against its complexity having been raised in the thirteenth
century by Alphonso X. , king of Castile, the author of the astronomical ' Tables '
associated with his name (published during 1248-1252).

INERTIAL SYSTEM OF REFERENCE 5

privilege of reference-system was taken away from our earth and
transferred to the sun on the ground of purely kinematical con-
siderations of simplicity, a few centuries before Newton. But after-
wards the Copernican or the ' fixed-stars ' system of reference appeared
to be wonderfully appropriate to Newtonian mechanics, both in its
original shape and in its later (chiefly formal) development by
Laplace for celestial and by Lagrange for terrestrial and general
problems. It soon became the final reference-system of mechanics.
It is relatively to this ' fixed-stars ' system of reference that the law of
inertia has proved to be valid. We will call it, therefore, following
the modern habit, the inertial system, or sometimes, also, the New-
tonian system of reference* It is relatively to this system that spin-
ning bodies behave in the characteristically simple manner which has
led many authors to speak of their property of 'absolute orientation.'
Or, to put it in less obscure words, it is relatively to the inertial system
that the vector called angular momentum is preserved, both in size
and in direction, — this property being a consequence of the funda-
mental laws of Newton's mechanics, and, at the same time, a perfect
and most instructive analogue to Newton's First Law of motion.!
The most immediate and tangible manifestation of this property is
that the axis of a free gyroscope (practically coinciding in direction
with its angular momentum) points always towards the same fixed
star ; thus having the simplest relation to the inertial system, since it
is invariably orientated in this system of reference. Notice that it
would, therefore, be more extravagant to say that the axis of such a
gyroscope moves relatively to the earth than vice versa, — though
apparently, bodily, the gyroscope of human make is such an incon-
spicuous tiny thing in comparison with our planet. The conservation
of the angular momentum, or moment of momentum, 2wVrv,| of
the whole solar system, which is best known in connexion with
Laplace's ' invariable plane,' is but the same thing on a larger scale
than that exhibited by our spinning tops. But this only by the

* We speak of it in the singular, instead of infinite plural, only for the sake of
shortness. For, as is well known, if S, say the * fixed ' stars, be such a system,
then any other system 2' having relatively to S any motion of uniform (rectilinear)
translation is equally good for all purposes.

tThis point is expressly insisted upon and successfully applied to didactic
purposes in Professor A. M. Worthington's Dynamics of Rotation, sixth edition,
new impression 1910 ; Longmans, Green & Co., London.

J See, for example, the author's Vectorial Mechanics, Chap. III. ; Macmillan &
Co., London, 1913.

6 THE THEORY OF RELATIVITY

way. What mainly concerns us here is that the ' fixed-stars ' system
— or, more rigorously, any one out of the oo 3 multitude of equivalent
inertial systems — has gradually turned out to be peculiarly fitted as
a system of reference for the representation of the motion of material
bodies.

But also with this system of reference the laws of motion have
their simple, Newtonian form only for a / measured in a certain way,
i.e. for a certain clock or time-keeper, e.g. approximately the earth in
its diurnal rotation, or, more exactly (in connexion with what is
known as the frictional retarding effect of the tides), a time-keeper
slightly different from the rotating earth. This is equivalent to
defining as equal intervals of time those in which a body not acted on
by ' external forces,' i.e. very distant from other bodies or otherwise
suspected sources of disturbance, describes equal paths.* In main-
taining the motion of such and such a body in such and such circum-
stances to be uniform, we do not make a statement, but rather are
defining what we strictly mean by equal intervals of time. Selecting
quite at random a different time-keeper, we could not, of course,
expect the same simple laws to hold, with respect to the inertial
system of reference. But with another space-framework of reference
another time-keeper might do as well.

Thus we see that, to a certain extent, the choice of a system of
reference in space has to be made in conjunction with the selection
of a time-keeper. Our x, y, z, /, the whole tetrad, the space and time
framework must be selected as one whole. That kind of 'union'
emphasized by the late Hermann Minkowski, the joint selection of
x-> y-> z> t> manifesting itself in the modern relativistic theory by the
consideration of a four-dimensional 'world' (instead of time and
space, separately), is not altogether such an entirely new and revolu-
tionary idea as is generally believed ; for to a certain extent, and in a
somewhat different sense, it is as well a requirement of Newtonian
mechanics, and, more generally, of the classical kind of Physics, as of
modern Relativity. What difference there really is between the two
we shall see in the following chapters.

* Thus it is manifest that the science of mechanics does not describe the motion
of bodies in its quantitative dependence upon ' time, flowing at a constant rate '
{Newton), but literally gives only sets of simultaneous states of motion of the
various bodies, the time-keeper itself being included. What is besides contained
in these sets or successions is a non-quantitative element, namely, of what is
vaguely called 'before' and 'after.'

CHOICE OF TIME-KEEPER 7

Meanwhile we have touched, in passing, the fourth variable /,
and this brings us to our second point, namely, the definition of
physical time, the selection of 'the independent variable /' of our
physico-mathematical equations, but viewed more generally, and more
carefully, than above, where we have touched it only incidentally.

To explain this question, of capital importance for almost every
quantitative physical research, I must ask you to direct your attention
to the following considerations.

Suppose we do not limit ourselves to the investigation of motion
only, but are concerned with every possible kind of physical pheno-
mena, such as conduction of heat or electricity, diffusion of liquids or
gases, melting of ice, evaporation of a liquid, etc., etc., and that we
propose to describe the progress of these phenomena in time, to trace
their history, past and future. How are we, then, to select our time-
quantity /?

First of all, we cannot, of course, take it to be Newton's * absolute
time,' which is defined, according to a quotation from Maxwell,* as
follows :

'Absolute, true, and mathematical Time is conceived by Newton
as flowing at a constant rate, unaffected by the speed or slowness of
the motions of material things. It is also called Duration.'

For, supposing there is such a thing,! we do not know how to
find or to construct a clock which measures this 'absolute time,'
even approximately; that is to say, we have no criterion to distinguish
such a clock from a 'wrong' one. And thus, certainly, we cannot
use this kind of definition for physical purposes. How are we then
to measure our /? Granting that the selection of a chronometer
indicating our t is (at least within certain wide limits) arbitrary or
free, what is the requirement on which we have to base our choice ?

Now, it seems to me that the first and most general requirement,
which may also be seen to be tacitly assumed in all the investigations
of both the more recent and classical natural philosophers, especially
physicists and astronomers, is

that our differential equations, representing the laws of physical
(and other) phenomena, should not contain the time, the variable /,
explicitly,

* Matter and Motion, page 19.

t But, as a matter of fact, the phrase ' flowing at a constant rate ' is simply
meaningless.

8 THE THEORY OF RELATIVITY

i.e. that for any sufficiently comprehensive physical system, of
which the instantaneous state is defined, say, by /lf /2> •••Ai» tne
differential equations should be of the form

-^7 =/t(A» A>---A)> i /Ax

I

J

2=1, 2, ... «.

This requirement is also intimately connected with a certain form
of what Maxwell* calls 'the General Maxim of Physical Science"
and what is commonly called the Principle of Causality.

To make my above statement more intelligible to a wider circle of
(non-mathematical) readers, let us consider s6me very simple examples
which will enable us also to see the exact meaning of instantaneous
' state' of a system and to learn to distinguish between two very
important and large classes of systems: i) complete or [undisturbed^
and 2) incomplete or ''disturbed' systems.

Suppose we have a small metallic sphere,! suspended somewhere
in a large dark cellar kept at constant temperature a, receiving no
heat, radiant or other, from without. Suppose we heated the sphere
to 100° C, which is to be >a (say, a = o°C), and from that instant
left it to its own fate. We return to it after an hour, as measured,
say, on one of our common clocks (i.e. rotating earth as time-keeper ),
and we find it has cooled down, say, to 90°. Thus :

/ 0

/0 100°, 1 .'. A0= - 10°,

/0 + i h. 90°. J for A/= i h.

Now, if we repeated the whole experiment to-morrow or next week,
we should find that during one hour the fall of temperature of our
suspended sphere would again be from 100° to 90°, i.e. A#= - 10°
for A/= ih. We could make similar observations for any other stage
of the cooling process of our little sphere (say down from 50° instead
of 100°) and for other time-intervals (say Jh. instead of i h.), arbitrarily
small, J and, repeating our observations, we should find again and again
the same permanency of results, — only with different values of A 6 for
different intervals A / and for different starting temperatures.

* Matter and Motion, p. 20, first paragraph of Art. xix. ; see also p. 21, lines t]-\\.

•\' Small' only so as not to be obliged to consider the different temperatures
of its various parts.

£ Or practically so, at least.

COMPLETE AND INCOMPLETE SYSTEMS 9

Thus, the temperature 6 of our sphere, placed in the specified
conditions of its environment, varies with time (ordinary clock-time)
in a certain determinate way, namely, so that starting from a given
temperature 0, its change during a given time-interval A/=/2-/'1, is
always one and the same, that is to say, no matter when this happens,
independently of /^ /2, but depending only on

Now, such a system, i.e. the sphere in its above environment, I
propose to call an undisturbed or, what for the beginning is more
cautious, complete system. And, in this case 0 being the only-
quantity on whose instantaneous value the whole (thermal) future
history of our sphere depends, we shall say, in accordance with
general use, that the instantaneous value of the temperature 0 defines
the instantaneous state of our system (a being supposed given once
and for ever). In the case before us we have a one-dimensional
system, which may be called also a system of one degree of freedom.*

Take the limit of the mean rate of change A0/A/ for A/->o ; then
the differential equation of our simple system will be of the form

S-yw, «>

which may be read : the instantaneous time-rate of change of the
temperature is a function of its instantaneous value only.f We
know in this case that f(Q)= —h(Q — a) approximately, when 6 — a
is small, where h is a positive constant ; but the particular
form of the function / is for our present purposes a matter of
indifference.

Let us, on the other hand, consider a similar sphere suspended,
say, in a window, exposed south, in a land in which the sun is wont
to shine often. Then, for the same starting value 0 and same A/,
the change A 6 will be different at different times of the day, e.g. larger
from 7 till 8 a.m. than from 2 till 3 p.m., larger in winter than in
summer, and so on. Now, a system such as this sphere we will call
a disturbed system or a system * exposed to external agents,' or better
an incomplete system, for this concept does not presuppose the know-
ledge of what is meant by 'action' of one system upon another.

* Observe that n mechanical • degrees of freedom ' amount to 2n degrees of
freedom in the sense here adopted.
t See Note 1 at the end of the chapter.

io THE THEORY OF RELATIVITY

In the present case the differential equation of our system will be
of the form

7/3

§-*•(*, /), (a)

/ being again measured with the ordinary (earth-)clock, and g being
some function involving / in a very complicated manner.

Now, according to the above general requirement, our /-clock
would be the right one, the peculiarly fitted one, for our first
physical system, (i), but not for the second, (2).

By selecting a different time-keeper we might possibly convert
some (not all) ' disturbed ' into ' undisturbed ' or complete systems ;
but then we should spoil the completeness of (i). Let us see, first
of all, what other clocks we can take instead of our original one
without spoiling the simple property of (i). Instead of /, take

then (i) will be transformed into

Thus, if the property of completeness is to be preserved, <£(/) must
be a constant, and consequently T a linear function of /, say

amounting only to a different initial point of time-reckoning and
to the choice of a different time unit.

Now (2), the equation of our second sphere, is not of the form
d6jdt=^(t).f(B\ but rather of the form

7/3

£-/[f- •(*)]+ <?<4;

consequently, if we wished also to sacrifice the completeness of (i),
we certainly cannot transform (2) into an undisturbed or complete
system, by any T=<j>(t). Hence the moral: certain incomplete
systems cannot be made complete by merely selecting a new clock
instead of the old one, and such systems I propose to call essentially
incomplete systems.

But suppose we had a system obeying a law of the form

SYSTEMS MADE COMPLETE n

i.e. a sphere as in (i), but having a coefficient h (coefficient of what
Fourier called external conduction, divided by specific thermal
capacity), which due to some visible changes of the sphere's
surface, such as oxidation, is variable^ instead of being constant.
Then we could represent it as a complete system by taking instead
of the /-clock another clock indicating the time

T=\h(f)dt, say = ^(/);
Jo

but, F(f) not being a linear function of the old time, this innovation
would at once spoil the completeness of (i).

At this stage we would find ourselves in face of an alternative :
which of the two systems, (i) or (3), is to be saved, which is to
be sacrificed ? And, correspondingly : which of the two clocks, the
/-clock or the T^clock is to be selected as time-keeper? If we
could detect no differences between the spheres (i), (3) — besides
that of their respective thermal histories — the choice would be
difficult, or rather arbitrary, quite a matter of taste or caprice. But,
say, the latter sphere, (3), gets oxidized, shrinks or expands, and
what not, and the former, (i), remains sensibly unaffected by the
process of repeated cooling and heating. Therefore, following the
maxim or principle of causality, we would conserve our /-clock,
best fitted for (i), and would try to convert (3) into a complete
system in a different way, namely, by taking account explicitly of the
oxidation of the sphere's surface, of its dilatation, and so on, i.e.
by introducting besides 6 other quantities, say, the amount m of
free oxygen present in the enclosure and the radius r of the sphere,
and by defining the state of the system by the instantaneous values
of 0, m r.

In this way, retaining our old clock, we should have converted the
originally disturbed system of one degree of freedom into a complete
system of three or more degrees of freedom. As a rule, we do not
reject our traditional time-keeper at once. Encountering an incom-
plete or disturbed system, every physicist will, first of all, try to throw
the ' disturbances ' on some ' external agent ' rather than on his clock.
He will look round him for external agents, almost instinctively
following the voice -of the maxim of causality, whispering to him, as
Maxwell puts it (Matter and Motion, p. 2 1) : ' The difference between
one event and another does not depend on the mere difference of the
times.' And finding nothing particularly suspect in the nearest

12 THE THEORY OF RELATIVITY

neighbourhood, he will look farther round, or deeper into, the system
in question.

Similarly, if we amplified the system of our second example (the
sphere cooling before an open window), taking in the sun varying in
position, the atmosphere, and possibly a host of other things, we
would obtain a larger, more comprehensive system which, though
more complicated than the original one, would satisfy us as being
undisturbed, with our old /-clock.

So it is in many other cases. Thus, we can say :

Adding to a given fragment of nature (system), which in terms
of a certain /-clock behaves like a disturbed or incomplete system
(/D/2> •••At)> *•*• obeys the equations

,...A, 0, (4)

* = I, 2, ... »,

fresh fragments of nature (with the corresponding parameters
Pn+\ i - • - Pn+m)i we often obtain a new, larger,* system which,
still with the same /, is undisturbed or complete :

i= i, 2, ... n + m.

In short, we complete the system Sn to Sn+m. The /, implied
here, is practically the time indicated by that clock which proved
peculiarly fitted for the description of our previous stock of experi-
ence. Thus, for example, Fourier's theory of conduction of heat
was preceded by the triumphs of classical mechanics ; and if asked
what the / in his fundamental equation

meant, he would, doubtless, answer that it is to be measured by the
rotating earth as time-keeper, though he hardly ever stopped in his
researches to consider this matter explicitly.

Thus, generally, we do not reform our traditional clocks, but we
make our systems complete as in (5), by amplifying them. But

* Not necessarily larger in volume ; for often we introduce new parameters by
going deeper into the original system itself, sometimes as deep as the molecular,
atomic or even sub-atomic structure, say, of a piece of matter ; or being originally
concerned with the thermic history only, we supplement the temperature by the
pressure, volume, electric potential, and so on.

AMPLIFIED SYSTEMS 13

sometimes, when we think that we have made our system Sn+m
sufficiently comprehensive, that we have exhausted all reasonably
suspected material as possible 'external agents,' and when Sn+m
nevertheless continues to behave as an incomplete system, i.e. when
still

then, to make it finally complete, we decide ourselves to change
our /, our traditional clock, — especially if the change required is a
slight one. This procedure, of course, is possible only when the F^
in (6) are all of the form

Otherwise, we feel obliged to help the matter by introducing yet fresh
parameters /n+m+1, A+m+2> etc., and not finding real (perceivable)
supplementary material round us, we introduce fictitious supplements,
which sometimes turn out to be real afterwards, thus leading to new
discoveries.

From this it is also manifest that the Principle of Causality has the
true character of a maxim ; though of inestimable value both in
science and in everyday life, it is not a law of nature, but rather
a maxim of the naturalist.

We have classical examples of both the procedures sketched above,
viz. of reforming our clocks and of supplementing or amplifying a
system with the view of securing its completeness. In the first place,
to get rid of one of the inequalities in the motion of the moon round
the earth, astronomers have had recourse to the supposition that there
is a gradual slackening in the speed of the earth's rotation. Of
course, they did it in connexion with the tides and with immediate
regard to the fundamental principles of mechanics, implying also the
law of gravitation. But at any rate, in doing so, and in declaring that
the earth as a clock is losing at the rate of 8-3, or (according to
another estimate) of 22 seconds per century, they gave up the earth
as their time-keeper and substituted for the sidereal time / a certain
function T=$(t\ slightly differing from /*, as their new Akinetic
time,' as Prof. Love calls it.* Secondly, as is widely known, the
perturbations of the planet Uranus have led Adams and Le Verrier

*A. E. H. Love, Theoretical Mechanics \ second edition, Cambridge, 1906,
page 358. In connexion with our subject, the whole of Chapter XL of Prof.
Love's book may be warmly recommended to the reader.

14 THE THEORY OF RELATIVITY

(working independently) to complete the system by a celestial body,
at first fictitious, but then, thanks to admirable calculations based on

the -g-law, actually discovered and called Neptune. Notice that

both kinds of procedure have essentially the character of successive
approximations.

Any future researches of mechanical, thermal, electromagnetic
and other phenomena, either new or old ones but treated with
increasing accuracy, if leading to 'disturbed' systems, obstinately
withstanding the supplementing procedure (i.e. that consisting in the
introduction of fresh parameters pn+\> etc.), may oblige us to reform
also the newer, slightly corrected earth-clock, to give up the * kinetic
time ' of modern astronomy for a better one, more exactly fitted for
the representation of a larger field of phenomena, and so on by
successive approximation. It may well happen that we shall have to
give up the kinetic time for the sake of the ' electromagnetic time,' —
if I may so call the variable / entering in Maxwell's differential
equations of the electromagnetic field.* For suppose for a moment
that some future experimental investigations of high precision were to
prove that the variable / in

3E 9M

^- = c . curl M, -^- r = - c . curl E

ot ot

is not proportional to the kinetic time ; then the electricians would
hardly give up these admirably simple and comprehensive equations ;
they would rather sacrifice the kinetic time. Thus, in the struggle
for completeness of our physical universe, we shall have always to
balance the mathematical theory of one of its fragments, or sides,
against that of another. A great help in this struggle is to us the
circumstance that, though, rigorously, all parts of what is called the
universe interact with one another, yet we are not obliged to treat at
once the whole universe, but can isolate from it relatively simple

* Thus we read in Painleve's article (loc. cit. page 91): 'La duree d'une
ondulation lumineuse correspondant a une radiation determined (ou quelque duree
deduite d'un phenomene electrique constant} sera vraisemblablement la prochaine
unite de temps. ' This idea seems to be suggested first by Maxwell ; the cor-
responding wave-length would at the same time be the standard of length, when
the platinum ' metre ttalon ' will be given up. Thus it may happen that the
'kinetic length' (i.e. that based on our notion of a 'rigid' body) will be sacrificed
for the benefit of an optical or ' electromagnetic length ' in the same way as the
'kinetic time' may be replaced by an 'electromagnetic time.'

NEWTON'S EQUATIONS 15

parts or fragments, which behave sensibly as complete systems, or
are easily converted into such.

Herewith I hope to have explained to you, at least in its
fundamental points, the question of selection of a time-keeper.

Thus, we know, essentially, how to measure our /, at least in or
round a given place (taken relatively to a certain space-framework).
We do not yet know what is the precise meaning of simultaneous
events occurring in places distant from one another. But the notion
of simultaneity, especially for systems moving relatively to one
another, belongs to the modern Theory of Relativity, and is, in fact,
a characteristic point in Einstein's reasoning. Therefore it will best
be postponed until we come to treat of the principal subject of this
volume.

We could now pass immediately to the history of the electro-
magnetic origin of the modern principle of relativity, extending from
Maxwell to Lorentz. But since we already have come to touch,
more than once, Newtonian or classical mechanics, let us dwell here
another moment upon this subject.

Let us call 2 one of the ' inertial ' systems of reference, say the
system of 'fixed' stars, and let xit yit zt be the rectangular co-
ordinates of the z'-th particle* of a material system, relatively to 2,
at the instant /. Then the Newtonian equations of motion are

d~X{ „ 1 0 »

mi~dfl= *' *' ' '

or

dxi _ dyi _ dzi _
~di~U" ~di~V" ~di = W"

dt l dt dt

where mit the masses, are constant scalars belonging to the individual
particles, / is the 'kinetic time' and Xit etc., are functions of the
instantaneous state of the material system, i.e. of the instantaneous
configuration and (in the most general case) of the instantaneous
velocities of the particles relatively to one another, which for .certain
systems may, but for a sufficiently comprehensive system do not,
contain explicitly the time /. If the material system is subject to

constraints, say

m = o, Y = o» etc.,

* The material ' particle ' may also play the part of a planet or of the sun, as in
celestial mechanics.

1 6 THE THEORY OF RELATIVITY

then X{, etc., contain, besides the components of what are called the
impressed forces, also terms like

which depend only upon the relative positions and relative velocities
of the parts of the system (i.e. of the mass-particles) to one another or
to the surfaces or lines on which they are constrained to remain,
or to the points of support or suspension entering in such constraints.
Thus the bob of a pendulum is constrained to remain at a constant
distance relatively to the point of suspension, the friction of a body
moving on a rough surface depends on its velocity relative to that
surface, and so on. Consequently, if instead of 2 any other system
of reference 2'(#', y'9 z) is taken, having relatively to 2 a purely
translational, uniform, rectilinear motion, Xi, YI, Zi are not changed.
And the same thing is true of the left-hand sides of the equations of
motion. For, if #/, etc., be the coordinates of the z-th particle
relatively to 2' at the instant t, and if we take, for simplicity, the
axes of x ', y, z parallel to and concurrent with those of x, y, z
respectively, then

. .

t =t,

where (u, v, w) is the constant velocity of 1! relatively to 2, and
where the fourth equation is added to emphasize that the old time t
is retained in the transformation. Consequently,

, dxl dXi
Ui = ~jfr = -jj--u = Ui-u, etc.

(and for any pair of particles ul - «/ = ut - ujt etc.), and
dul _dui dvl _dVi dwl dwi

W = ~dt' W = ~di' ~W = ~di*

which proves the statement.

Thus, the equations of motion (8), or, in vector form,

remain unchanged by the transformation (9), or, written vectorially,
by the transformation

NEWTONIAN TRANSFORMATION 17

where v, the resultant of the above u, v, w, is the vector-velocity
of 2' relatively to 2. As regards the time, we could write also
t' = at + b (a, b being constants), but this would amount only to a
change of units and shifting of the beginning of time-reckoning.

In view of the above property, the linear transformation (9) or (90),
v being any constant vector, is called the Newtonian (and by some
authors the Galileian) transformation. Thus we can say, shortly :

The equations of classical mechanics are invariant with respect
to the Newtonian transformation. J if ^^ &^

Notice that v being quite arbitrary, both as regards its size (or
tensor) and direction, we have in (90) a manifold of oo3 transforma-
tions, and all of these form a group of transformations. For, if

r^ri-Vj/; /' = /,
and

r/' = r/-v./; /" = /',
then

r/' = ri-v/; /" = /',
where

v = V! + v2. (10)

We shall refer sometimes to (9) or (90) as the Newtonian group.

Notice the simple additive property (10), to be compared later on
with a less simple property of the corresponding group in modern
Relativity.

Thus, there is no unique frame of reference for classical mechanics ;
if the Newtonian equations of motion are strictly valid relatively
to the framework 2 of the ' fixed ' stars, they are equally valid
relatively to any other out of the oo3 frameworks 2', connected with
2 by (9), say relatively to the solar-system frame, which has relatively
to 2 a uniform velocity of something like 25 kilometres per second,
towards the constellation of Hercules.* Therefore, by purely in-
ternal mechanical experiment and observation, i.e. not looking out-
side to external systems, we could never distinguish the solar frame
2' from 2, that is to say, 2', like 2, does not show any anisotropy
with regard to mechanical phenomena. The same remark applies, •
with sufficient approximation, to the earth's annual motion : it is not
ascertainable by purely terrestrial mechanical experiments.

Physicists hoped to detect this motion which they called also
* the motion relative to the aether,' by the means of purely terrestrial

* Quoted after Painleve, loc. cit. page 117.
S.R. B

1 8 THE THEORY OF RELATIVITY

optical or electromagnetic experiments, — we shall see later how
unsuccessfully.

In other words, seeing that there is no unique 'kinetic' space-
framework, they tried to find a unique ' optical ' or ' electromagnetic '
reference-system, the 'aether,' or rather to show that this wonderful
medium, already invented for other purposes, was such a unique
frame of reference. But the results of all experiments of this kind
have been obstinately negative.

It is chiefly this which has led to the construction of the new
theory of relativity.

NOTES TO CHAPTER L

Note 1 (to page 9). To show, generally, the connexion between
the integral form of the properties of a complete system, as stated in the
above illustrations, and its differential form, of which eq. (i) is an
example, let us consider such a system of n degrees of freedom. Let its
state at any instant / be determined by

Then, /0 = o being any other, say, past instant,

A(0=^[A(o),-A(o); 4 *»i,2,,,.«,

where Pi is a symbol of an operation or a function, implying besides the
'initial' state p(o) the time-interval t=t-tQ elapsed, but independent of
the choice of the initial instant. This is the finite or integral way of
expressing that the system is complete. Now let t=a be any particular
instant and t=c another instant of time, such that

c—a + l).
Then

4

so that the transformations corresponding to the passage of the system
from any of its states to its successive states form a group of transforma-
tions, t being the (only) 'parameter' of the group. Thus we can imitate
Lie's general proof of his Theorem 3 (Sophus Lie, Theorie der Transfor-
mationsgruppen, Leipzig, 1888 ; Vol. I.) for this simplest case of one

COMPLETE SYSTEM 19

parameter. Considering ^(o), ...pn(o\ a, c as independent variables,
differentiate pi(c) with respect to a ; then

t(tf) da ^n(a} da 'db 'da

%w

but 3£/dtf= - i ; therefore

) PM*

da

/=!, 2, ...«.

Now pi(c\ ...pn(c) are mutually independent ; otherwise less than n
quantities p would suffice for the determination of the state of the system,
contrary to the supposition. Therefore the functional determinant

does not vanish identically, and the above system of n equations can be
solved with respect to dp^(a)\da^ etc., leading to

*->«[AC*i -AW; *1 '=', v~*

But these equations must be valid for all values of the mutually in-
dependent magnitudes b and a. Giving therefore to b any constant
value, and writing t instead of #, we obtain for any /,

and this is the differential form alluded to, flt /2, ... fn being functions
of the instantaneous state only.

It is instructive to consider the instantaneous state of a system as a
point in the w-dimensional space, or domain of states 5n, (/i, pz ---pn\
and to trace in this ' space ' the lines of states, i.e. the linear continua
of states assumed successively by different copies (exemplars) of the
system, starting from given initial states. The differential equations of
these lines of states, or, as Lie calls them, the 'paths (Bahncurven} of
the corresponding infinitesimal transformation,' are

_= =

A A fn '

A complete system may then be characterized by saying that the lines
of states are fixed in the corresponding space 5re, like the lines of flow of
an incompressible fluid in steady motion. A copy of the system, or
rather its representative point, placed on one of these lines remains on

20 THE THEORY OF RELATIVITY

it, moving along it in a determined sense. (For particulars of physical
application of these concepts, see the author's paper in Ostwald's
Annalen d. Naturphilosophie, Vol. II. pp. 201-254.)

Note 2 (to page 12). Systems obeying partial differential equations,
as for instance that of Fourier,

_

adduced in the text, may be considered as systems of infinite degrees of
freedom. The instantaneous state of such a system implies an infinite
number of data^-, or say p=p(x,y, z\ given as a function of _r, j, z for
every point of a portion of space coextensive with the system, as for
example the instantaneous temperature for every point of a cooling body
of finite dimensions, in which case the system will have co3 degrees of
freedom. Instead of one we may have also two or more functions of
x, y, z, defining the instantaneous state, as for example two vectors,
amounting to six scalars, for an electromagnetic system (field), the
differential equations being in this case those of Maxwell,

c>E .__ 3M lt,

-~-- = £.curlM, -~TT- = - c . curl E.

Here, as in the above example, the right-hand sides do not contain the
time explicitly, but depend only on the space-distribution of magnitudes
referring to the instantaneous state. If such be the differential equations
and if also the limit or surface-conditions do not contain the variable /
explicitly, the system of infinite degrees of freedom will be a complete or
undisturbed one, in the sense of the word adopted throughout the chapter.
Thus a heat-conducting sphere, of finite radius R, obeying in its interior
Fourier's equation and whose surface is thermally isolated or radiates
heat into free space, will be a complete system ; for its boundary con-
ditions, viz.

30

Wr=°
or

~j- = const, x (9- const.)

respectively, do not contain the time explicitly. But a sphere (like the
earth), whose surface is kept at a generally variable temperature by
means of external sources (like the sun), will be an incomplete system,
unless we amplify it by taking in the ' sources ' themselves.

CHAPTER II.

MAXWELLIAN EQUATIONS FOR MOVING MEDIA AND
FRESNEL'S DRAGGING COEFFICIENT. LORENTZ'S
EQUATIONS.

THE modern principle of relativity arose on the ground of Lorentz's
electrodynamics and optics of moving bodies. Einstein's work, in
fact, consisted mainly in deducing logically, on the basis of plausible
and sufficiently general considerations, certain formulae of space and
time transformation, which in Lorentz's theory had partly a purely
mathematical meaning and partly the character of an hypothesis
invented ad hoc (' local time ' and the contraction hypothesis,
respectively). In a word, Einstein has given a plausible support to,
and a different interpretation of, what appeared already in the theory
of the great Dutch physicist. In its turn, the theory of Lorentz,
based on the macroscopic treatment of a crowd of electrons (though
later supported and made vital by physical evidence of an entirely
different kind), was constructed by its author chiefly with the purpose
of accounting for optical phenomena in moving bodies, which may be
best grouped summarily under the head of Fresnel's * dragging coeffi-
cient ' and with which the equations of Maxwell and of Hertz-
Heaviside have proved to be in complete disagreement.

Now, it seems to me that the best, most natural and most efficient
way of propagating new ideas (if indeed there is such a thing arising in
the collective mind of humanity) is to show their intimate connexion
with older ones, and the more so when the new ideas have the
reputation, widespread but partly unjustified in our case, of being of
a very revolutionary character. It will be advisable, therefore, before
entering upon our proper subject, to turn back to Lorentz and
Maxwell. In doing so, I must warn the reader at the outset that the
new Relativity, though grown on electromagnetic soil, does not — in
spite of a current opinion — require us at all to adopt an electro-

22 THE THEORY OF RELATIVITY

magnetic view of all natural phenomena. Nor does it force upon us
a purely mechanistic view, which till recently held the field, before
the pan-electric tendencies arose. Modern Relativity is broader
than this : it subordinates mechanical, electromagnetic and other
images to a much wider Principle which is colourless, as it
were.

Thus, the reason of returning here to Maxwell is, in the first place,
of an historical (and partly didactic) character. But we have yet
another reason for dwelling in the present chapter upon the great
inheritance left to Science by Clerk Maxwell. It is widely known
that but a few things of the old system of physics have remained
untouched by the modern principle of relativity, though the changes
required are generally but very slight. In fact, almost nothing of the
old structure has been spared by the new theory of relativity ; but
Maxwell's fundamental equations, namely those known as his
equations for * stationary •' media, have been spared. More than
this : not only have they been preserved entirely in their original
form, without the slightest modification of any order of magnitude
whatever, but they form one and the best secured of the actual
possessions of the new theory, the largest and brightest patch of
colour, as it were, on the vast and as yet mostly colourless canvas
contained within the frame of the new Principle. Moreover, a
peculiar union or combination of the electric and magnetic vectors
which appear in Maxwell's equations of the electromagnetic field
became the standard and prototype (not as regards physical meaning,
but mathematical transformational properties) of a very important
class of entities admitted by the new theory (the so-called 'world-
six-vectors ' or * physical bi vectors ').

So much to justify the insertion of the following topics of the
present chapter.

Maxwell's fundamental laws of the electromagnetic field in a
* fixed ' or * stationary ' non-conducting dielectric medium * may be
expressed in integral form as follows :

I. Electric displacement-current through any surface <r bounded
by the circuit s = c x line integral of magnetic force M round s.

II. Magnetic current through <r = -rxline integral of electric
force E round s,

* Practically, fixed with respect to the earth, or, if not, then with respect to a
definite system of reference S, to be ascertained on further examination.

MAXWELL'S EQUATIONS 23

i.e. in mathematical symbols :

ii.

where (£, Jft denote the dielectric displacement or polarization and
the magnetic induction respectively, c a scalar constant, the velocity
of light in vacuum, n a unit vector normal to o-, the sense of the
integration round s, of which ^s is a vectorial element, being clock-
wise for a spectator looking along n (see Fig. i). Here, as throughout

FIG. i.

the volume, ((£n), etc., generally (AB), in round parentheses, denote
the scalar product of a pair of vectors :

(AB) = ^cos(A, B),

A, B being the sizes or absolute values of the vectors A, B.* Thus,
the surface element do- being considered as an ordinary scalar, the

surface integral I ((£n)dcr stands for the total number of Faraday tubes

(unit tubes) crossing o-, and the surface integral in n. has a similar
meaning with respect to the tubes of magnetic induction.

* If it were only for purely vectorial algebra and analysis, we could write, after
Heaviside, for the scalar product simply AB. But since we shall have to recur in the
sequel to Hamilton's quaternionic calculus, we reserve AB for the full quaternionic
product, and write therefore (AB) for the scalar product, i.e. for the negative
scalar part of the Hamiltonian product, and VAB for the vector product, thus

AB = S. AB + V. AB

= -(AB) + VAB.

24 THE THEORY OF RELATIVITY

Remembering the definition of ' curl ' by means of the line integral,
we may write I. and n. at once in differential form,

Sf-,c.,M,

or, in Cartesian expansion,

(la)

pf p

Every point or surface element of <r being fixed, relatively to the
system of coordinates x, y, z, round d's have been written on
the left hand to express partial differentiations with respect to /,
i.e. local time-rates of change of the corresponding vectors.

(la) or (i) is the Hertz-Heaviside form of Maxwell's differential
equations, although, if I am not mistaken, Maxwell himself on one
occasion employed this form. At any rate, the Hertz-Heaviside
equations for a stationary medium differ only formally from the
equations of Maxwell as given in his monumental ' Treatise ' and
in several papers ; the auxiliary potentials being easily eliminated.

As regards the relations obtaining between (£, JE and E, M
respectively, it will be enough to remember here that the first pair
of vectors are linear functions of the second, say,

(g = ^E and Jft = /*M, (2)

where A", /A are in the general case, of crystalline bodies, symmetrical
or self-conjugate linear vector operators, which in the simplest case
of an isotropic medium degenerate into ordinary scalar coefficients,
the dielectric ' constant ' or the permittivity, and the magnetic
permeability or the inductivity, — to adopt Heaviside's nomenclature.*
Notice that, using the relations (2), K and //. being supposed
given, we have in (i) two vectorial equations of the first order for
two vectors, so that if the 'initial' state, say E0, M0, and eventually
the limit-conditions, be given, the whole history of the field, past

* As yet we have no need to touch upon the subject of conducting media.

MAXWELL'S EQUATIONS 25

and future, is uniquely determined, — though in most cases the
mathematician may have the greatest difficulties in finding it out.
The electromagnetic field, as far as it obeys these equations,
is at any rate a complete system in the sense of the word
previously explained. It will be noticed later that the funda-
mental equations of the electron theory do not possess this simple
property.

From i., ii. we see immediately that the total current, electric
or magnetic, through all possible surfaces a- bounded by one and
the same circuit (s), has the same value. Taking therefore a pair
of such surfaces crl, <r2, which together form a surface (o-), enclosing
completely a certain portion r of the medium, and inverting one
of the normals of the component surfaces (Fig. 2), so that the

n=n.

FIG. 2.

normal n is directed everywhere outwards (or everywhere inwards)
with respect to the enclosed space, we see that, for any closed
surface (<r),

I (£n)^cr, (Jttn)dr = const, in time,
J(o-) J(<r)

the second constant being everywhere equal to zero, by experience.
In other words, the total electric charge enclosed by (<r) does not
vary in time, its magnetic analogue being constantly non-existent.
The same property being valid for any volume T, and remembering
that ' div ' or divergence is defined as the surface integral of a vector
per unit of enclosed volume, we may write also, in differential form,

div (& = p = const,
div JE = const. = o ;

26 THE THEORY OF RELATIVITY

p is the volume density of (true) electricity. The second property
is commonly expressed by saying that the tubes of magnetic induc-
tion are always closed, or that Jft has a purely solenoidal distribution.
The invariability of both divergences may be seen with equal ease
from (i), remembering that the operations div and 3/9/ are com-
mutative, while div curl = o, identically.

Thus, the full system of Maxwell's equations for a stationary
dielectric, which we will put here together for future reference, is

(3)

-^.curlM

-r.curlE; divjtt = o

the equation

/o-div(g

being here considered as the definition of the density p of electric
charge. Notice, in passing, that the ' electric charges ' have been
driven to the background by the Maxwellian theory (especially as
propagated by Hertz, Heaviside and Emil Cohn), as rather secondary
derivate entities, but to return later with increased vigour and to
reacquire their dominant position, viz. as fundamental elements of
the electron theory.

We shall not stop here to consider the general Maxwellian ex-
pressions of energy, ponderomotive force and of the corresponding
stress.

In vacua, and practically also in air under ordinary conditions,

so that Maxwell's equations (3) become

3E

-~- = c . curl M
ot

^r- = - c . curl E
ot

div M = o,
to which in the present case may be added also

divE = o (4^

expressing the absence of electric charge. Notice in passing that
these equations are not invariant with respect to the Newtonian

(4)

MAXWELL'S EQUATIONS 27

transformation. The transformation which does preserve their form
is of a different kind, as will be seen later.

The independent variable t appearing in Maxwell's equations (4)
for empty space may be taken, provisionally at least, as far as
experience goes, to be the ordinary or the kinetic time. And as regards
the (or a) space-framework, with respect to which they are intended
to be rigorously valid, let us call it once and for ever the system S,
whatever it may be. If the reader wants to fix his ideas he may think
of S as the ' fixed-stars ' system ; but as yet we cannot and need not
discuss this point thoroughly, being forced by the very nature of the
question to postpone it to a later chapter. At first sight it might
seem that (4) are wholly independent of a space-frame of reference ;
for the curls and div's can be, and primarily are, defined in terms
of line integrals and surface integrals respectively, and thus depend
only upon the distributional peculiarities of the respective vector
fields. But this means only that the equations in question are
independent of the choice of axes (#, jr, z) within S, the only condi-
tion being that they must be immovable relatively to 6" ; in other
words, curl E, curl M are vectors as good as E, M themselves * and
div E, div M are true scalars like a volume, for instance. Notice,
however, that, on the left hand of the equations, 3/3 / is to be the
local time rate of change of E or M, i.e. the variation in a point P
kept fixed. Now, this would be altogether meaningless if it
is not explained with respect to what frame the point P is to
be fixed. It would not help us very much if somebody told
us that P is to be a fixed point of the field or of a Faraday
tube ; for we have no means of identifying such a point. The
truth of what has just been said may be seen even more im-
mediately from the integral form of Maxwell's equations, i. and n.,
where for the present case <£, JE are to be identified with E, M ;
for the circuit (s) is to be kept 'fixed,' i.e. fixed with respect to
something.! Therefore we necessarily want a frame of reference,
and call it S.

* The distinction of what are called axial and polar vectors does not concern us
here.

fin the more general case of a ponderable medium, say in a piece of glass, the
circuit (s) is, of course, to be fixed in the glass ; but this would not be enough : the
whole piece of glass, as will be explained presently, must not move in an arbitrary
manner relatively to some external frame or other, if the laws I., n. are to be
valid, whether the observer does or does not share its motion.

28 THE THEORY OF RELATIVITY

To see the property of the scalar constant c, eliminate, in the usual
way, E or M, employing their solenoidal properties ; then

?&-*•+• (5)

where <£ means E or M, or any one of their Cartesian components
£lt ... M3; hence, in the case of plane waves, for example,

and

/ being an arbitrary function of the linear argument. Thus c,
in round figures 3.io10 cm. sec.'1, is the velocity of propagation
in empty space, relatively to S, of transversal electromagnetic waves
or disturbances, their transversality being an immediate consequence
of the solenoidal conditions, which, in the present case, reduce to
'd£l/'dx = o, fdMll'dx = o. Henceforth c will be referred to shortly
as 'the velocity of light,' and sometimes as the 'critical' velocity.

What is properly called a wave is a non-stationary surface of discon-
tinuity of E, M themselves or of their derivatives, which is individually
recognizable as such and can be watched when moving about. It is
the velocity of motion of such a wave, normal to itself, which is
properly called the velocity of propagation, as distinguished from the
phase-velocity of a continuous train of disturbances. Now it may be
easily shown that c is precisely the value of this true velocity of pro-
pagation for any form of the wave, plane or not, the property belonging
to every surface element of the wave, considered separately. (See
Note i at the end of the chapter.)

Notice that this property is quite independent of the direction of
the wave normal, i.e. of its orientation with respect to any axes drawn
in S. In other words :

Maxwell's equations imply isotropic as well as uniform * propaga-
tion in empty space relatively to S, i.e. to that system in which
they are valid. There are no privileged places or directions for
the electromagnetic disturbances.

Thus a continuous train of spherical waves, with centre O, will
remain spherical for ever, which may be seen also from (5). For a

* By ' uniform ' we mean homogeneous or constant in space and invariable in
time, c being constant with respect to both.

ISOTROPIC PROPAGATION 29

particular integral of that equation, adaptable to any initial state
<t>0 = -f(r)t is <£ = -f(r ± ct\ r being the scalar distance measured

from O. Again — which is more satisfactory — if cr be at any instant
a spherical surface of transversal discontinuity or a proper electro-
magnetic wave, then, expanding (or shrinking) with time, it will
remain spherical for ever, with centre O coinciding always with
that of the original cr, fixed once and for ever with respect to the
frame S, — quite independently of whether and how the material
source was moving at the instant when it originated that wave.
Thus a * point-source ' (and notice that a physical source of any
shape or finite dimensions may be regarded as such, provided we
go away from it far enough) producing a solitary disturbance, say
a flash of light, at the instant /0, will originate a wave which always
will be spherical of radius

having its centre where the source was at the instant /0, no matter
whither it went afterwards or whence it came, or how swiftly it flashed
through that place.

We shall have to return to this argument, of capital importance,
more than once ; but meanwhile we must leave it.

As has been already remarked, Maxwell's equations for '•stationary '
dielectrics, i.e. I. and n. with their supplements as given together
with their differential form under (3), have not only survived the
general massacre, but have very substantially enriched the new theory.
In fact, both the most particular and simple equations (4) for the
vacuum and the more general ones, (3), for ponderable media have
been incorporated into the possessions of modern Relativity, the
former in a quite easy way by Einstein (1905), and the latter
in a less easy and very ingenious way by Minkowski (1907).
On the other hand, it is needless to tell here again about the
wide field of experience covered by these equations and about
their numerous and successful applications in proper Electro-
magnetism, to say nothing about the electromagnetic theory of light
which soon after its creation proved to be much superior to the
elastic theory.

Serious difficulties arose only in connexion with the electro-
dynamics, and more especially with the optics of moving media, a
long time before the dates just quoted.

30 THE THEORY OF RELATIVITY

There are two different sets of what are commonly called Max-
wellian equations for moving media : i° a system of equations which
may be gathered together from different chapters of Maxwell's
' Treatise,' and which we shall call shortly the equations of Maxwell,
though it can be reasonably doubted whether Maxwell himself would
consent to attribute to them general validity, especially with the
inclusion of optics; and 2° a system of equations which Hertz
obtained by a certain, apparently the most obvious, extension
of the meaning of the form i., 11., and which Heaviside, inde-
pendently, constructed by introducing into Maxwell's equations a
supplementary term dictated by reasons of electro-magnetic sym-
metry ; these are widely known as the Hertz-Heaviside equations
for moving bodies.

We shall use for i° and 2° the abbreviations (Mx), (HH).
Neither has been able to stand the test of experience. Though
contrary to the historical order, it will be more instructive to con-
sider first the latter and then the former system of equations.

Let us return to the semi-integral form of electromagnetic laws i.
and ii., given, in words and symbols, on pp. 22-23. These are valid
for a ponderable dielectric medium or body, stationary with respect
to our frame S, and for any surface a- which, together with its
bounding circuit s, is fixed in the body. Thus the surface o-,
through which the 'current' is to be taken, is itself fixed in S.
Now, what Hertz did in order to obtain the required extension,
was simply to suppose that i. and 11. are still valid for a body,
rigid or deformable, moving with respect to S in any arbitrary
manner, provided that the currents on the left-hand side of these
equations are taken through a surface composed always of the same
particles of the body, or— to put it shortly — through an individual o-,
together with its s. This gives for the current per unit area of <r,
instead of the local time-rate of change 9®/3/, if v be the velocity
of a particle relatively to S,

^ + vdiv@ + curlV<£v, (6)

and a similar expression for the magnetic current,* while the right-
hand sides of i., ii., containing only the instantaneous values of
line integrals, remain obviously unaffected by the Hertzian require-
ment. The distribution of JE being supposed solenoidal, as

*See Note 2.

HERTZ-HEAVISIDE EQUATIONS

previously, the second term in the above expression is absent in the
magnetic current. Thus, transferring the curl-terms of the currents
to the right-hand sides, we obtain the required equations

(HH)

Heaviside calls V(£v/<: the ' motional magnetic force ' and Vvjft/r
the 'motional electric force,' considering them as a kind of impressed
forces.

In what we have called Maxwell's equations, the former of these
' motional forces ' and the convection current pv are absent ; other-
wise they are as (HH) ; thus

= c . curl M

(MX)

The connexions between <£, Jft and E, M are as in (3), except
that K, fj. may undergo continuous variations due to the strain
of the material medium. Also, div,.|ft = o, as in (3). Notice, in
passing, that the first of (HH) gives

or

dp
dt

+ p div v = o,

where <//9/*#=3/>/d/-f(vV)p is the variation at an individual point of
the body. Now, div v being the cubic dilatation, per unit time and
per unit volume, the last equation may at once be written

where dr is an individual volume-element of the material medium,
i.e. an element composed always of the same particles. Thus the
charge pdr of any such element remains invariable, being attached to
it once and for ever. The charge, being preserved in quantity, moves

32 THE THEORY OF RELATIVITY

with the body. In this respect it behaves like the mass, according
to classical mechanics. As regards the equations (Mx), they must be
considered as referring to the particular case of an uncharged body ;
Maxwell happened not to consider explicitly charges in motion ;
otherwise he wo^ld doubtless have brought in the term pv.

Now, both of these systems of equations, (Mx) as well as (HH),
are in full disagreement with experience, especially with optical
experience, terrestrial and astronomical, i.e. with experiments on the
propagation of electromagnetic waves (light) in bodies moving
relatively to the observer, and also in bodies moving with the
observer and with his apparatus relatively to the source, say relatively
to a star. *

The equations in question have also been manifestly contradicted
by electromagnetic experiments properly so called, viz. those of
H. A. Wilson and of Roentgen and Eichenwald ; * but it will be
enough to consider here only the difficulties met with on optical
ground, the other deviations being of essentially the same character,
while the optical examples, quite conclusive by themselves, seem to
be very instructive.

Let me explain to you fully what this disaccordance consists in.

To take the simplest case possible, let the material medium or

tf

FIG 3.

body move as a whole with uniform translational velocity v with respect
to S, and let plane waves of light be propagated in it along the
positive direction of v (Fig. 3). If the unit-vector i be the wave
normal, concurrent with the propagation, then v = vi. Let fo' be the
scalar velocity of propagation of the waves, when the material medium

* H. A. Wilson, Phil. Trans., A. Vol. CCIV. p. 121 ; 1910.— W. C. Roentgen,
Berl. Sitzber., 1885; Wiedem. Ann., Vol. XXXV. 1888, and Vol. XL. 1890.—
A. Eichenwald, Ann. der Physik, Vol. XI. 1903.

THE DRAGGING COEFFICIENT 33

is stationary in S, and b their velocity of propagation, . as judged
from the ^-standpoint, when the medium is moving with its actual
velocity. What is the relation between b and b', v? If we were
concerned with waves of sound, instead of light waves, then b would
be simply the sum of b' and of the whole v; the waves would be
entirely dragged by the medium, say air or water, with its full
velocity. But the case before us is different. Write, generally,

or

then K, whatever its value, will be what is called the dragging coefficient,
indicating the fraction (if it happens not to be the whole) of the
medium's velocity conferred upon the waves. What is, then, the
dragging coefficient in the case of electromagnetic, and especially of
luminous waves?

According to (HH) it is, obviously, equal to unify. To see this
we have no need to integrate these differential equations,* but simply
to remember Hertz's interpretation of the laws i., n., which furnished
him with these equations (p. 30). For according to that interpreta-
tion, and extension, of i., n., the electromagnetic disturbances behave
relatively to the material medium (generally in each of its elements,
and in the present case, of rigid translation, throughout the whole
medium) just as if it were stationary. Hence, on the ground of
classical kinematics of course, the velocity of the medium is simply
added to that of the waves, precisely as in the case of sound. Thus,
K = i, according to (HH).

Let us now see what is the value of the dragging coefficient
according to (Mx). Take the simplest case of an isotropic medium ;
then

where, by the way, /* = i for light waves. Measuring x along i in the
system S, take E, M, and therefore also (£, JH, proportional to a
function of the argument ^-b^, so that b will be the velocity of

* Though the reader, to satisfy himself, may do so. Proceeding similarly as in
the case of (Mx), worked out in Note 3 at the end of this chapter, he will soon
find that b = ti' + v.

S.R. C

34 THE THEORY OF RELATIVITY

propagation relatively to S, as above, and by a simple calculation
(Note s)

or

b = *'(i+*«W* + ^, (7)

where /3 = v/c and where 72 = <r/V is the index of refraction of the
medium. Now, in all actual experiments, by means of which the
dragging of light can be determined, /? is a small fraction, viz. io~4
in the case of Airy's astronomical, and much smaller in that of
Fizeau's terrestrial experiment, both to be considered later. There-
fore terms of the order of /24 can certainly be rejected, so that

and

but here even the /3-term may be safely omitted, so that finally

Thus, we have for the dragging coefficient according to (HH)
and (Mx), respectively,

K=I, (HH)

K=K. (MX)

Now, both of these are radically wrong, the true one, i.e. that
showing excellent agreement with experiment, being Fresnel's widely
known dragging coefficient (coefficient d? entrainement)

•irtu j

K - i - — , (Frsnl)

/ /• ^"

• • -

where n is the index of refraction. It is, for more than one reason,
worth our while to dwell here upon the interesting history of
Fresnel's coefficient.

The phenomenon of stellar aberration, discovered by Bradley in
1728, found its immediate explanation when the assumption was
made that the light-waves do not share in the earth's orbital motion,

*This result was obtained by J. J. Thomson. See Heaviside's Electromagnetic
Theory, Vol. Ill, §471 et seq., where some interesting remarks regarding this and
allied subjects may be found.

ABERRATION 35

and, consequently, in the motion of the tube of the telescope
(if filled with air or empty). In fact, making this assumption, the
aberrational formula

and, for 0 = Tr/2, - $ °

- = sin </> = tan </>, (8a)

is easily obtained by using the widely known analogy of a ship in
motion pierced by a shot fired from a gun on the shore.

Formula (8) gave, from Bradley's observations (<f> = 2o"-44) and
from the known velocity v of the earth's motion (30 kilom. per second),
a value for c, the velocity of propagation of light, which agreed very

Earth

FIG. 4.

closely with that obtained by Romer in 1676 from observations of
the eclipse of Jupiter's satellites. Thus (8) was verified. To state
the bare facts, it would have been enough to say simply that the tube
of the telescope, or the air contained in it, does not carry with it the
light coming from the star, whatever it may consist in (corpuscles or
waves). But to make the statement more tangible, it has been said
that the 'corpuscles' or the 'aether,' respectively, do not share in
the telescope's motion. Whereas aberration was explained by its
discoverer in terms of the corpuscular theory (each corpuscle of light
corresponding then most immediately to the shot in the above
analogy), it was Young who first showed (1804) how it may be
explained on the wave-theory of light and on the hypothesis that the
aether ' pervades the substance of all material bodies with little or no
resistance, as freely perhaps as the wind passes through a grove of

36 THE THEORY OF RELATIVITY

trees.' * This picturesque analogy fitted altogether the case of air,
which behaves very nearly like a vacuum, but not glass or water, for
which the ' grove of trees ' had to be replaced by a rather dense
thicket. But at any rate the above words of Young hit very near the
truth.

To put it shortly, in the case of air" the dragging is «//, or nearly
so, K === o.

But the case is different for optically denser media, having, for
light of a given frequency, an index of refraction ;/, sensibly different
from unity. For if K were nil also for such media, we should have to
replace c in (8) by the smaller velocity of propagation cjn^ (so) tharTthe
angle of aberration ^vould be) different for optically different media,
whereas it has been proved experimentally to be just the same as in
the case of air. More generally, Arago concluded from his experi-
ments on the light of stars that the earth's motion has no sensible
influence on the refraction (and reflection) of the rays emitted by
these light-sources, i.e. that the rays coming from a star behave, say,
in the case of a prism or a slab of glass, precisely as they would if the
star were situated at the point in which it appears to us in conse-
quence of ordinary Bradleyan (air-telescope) aberration, and the earth
were at rest relatively to the star. Arago himself tried to explain this
result of his experiments on the corpuscular theory, and on the
supplementary hypothesis that the sources of light impress upon the
corpuscles an infinity of different velocities, and that out of these
none but those endowed with a certain velocity (±-oi %) have the
power of exciting our organ of sight. But this strange hypothesis
entangled him in a maze of difficulties, and the whole theory, not free
from other difficulties, does not seem to have satisfied its author.
At any rate, Arago proposed to Fresnel to investigate whether the
above result of his observations could not be more easily reconciled
with the wave theory of light.

It was in answer to this invitation that Fresnel wrote in 1818 his
celebrated letter to Arago * on the influence of the earth's motion
upon certain optical phenomena,'! in which he gives a beautiful

* Phil. Trans., 1804, p. 12, — as quoted by Whittaker in A History of the
Theories of Aether and Electricity -, p. 115 ; London, 1910.

t ' Lettre d'Augustin Fresnel a Francois Arago, sur 1'influence du mouvement
terrestre dans quelques phenomenes d'optique,' Annales de chim. et de phys.y
Vol. IX. p. 57, cahier de septembre, 1818; reprinted in Fresnel's CEuvres com-
pletes, Vol. II., Paris, 1868; No. XLIX. pp. 627-636.

FRESNEL'S DRAGGING COEFFICIENT 37

solution of the problem, and which has since become one of the most
solid supports of modern inquiry into the optics of moving media.
Here appears for the first time his 'coefficient d'entrainement,'
already mentioned above. Fresnel based the theory of aberration,
and associated matters, on the following hypothesis, which turned out
to be a very happy guess indeed :

Fresnel supposed that the excess, and only the excess, of the
aether contained in any ponderable body over that in an equal
volume of free space is carried along with the full velocity, v, of the
body ; while the rest of the aether within the space occupied by the
body, like the whole of the free aether outside, is stationary, — with
respect to the fixed stars, of course.

This amounts * to supposing that the velocity of propagation of the
light-waves is augmented only by the velocity of the ' centre of
gravity ' (centre of mass) of the whole mass of the aether contained
in the body. This velocity will, generally, be but a fraction of v.
Call it KV ; then K will be what has above been called the dragging
coefficient. Let pQ be the density of the aether outside the
body, and p its density within the body; then, by Fresnel's
hypothesis,

or

* = I ~ PO/P'

Now, e being the coefficient of elasticity of the aether within the
body, and eQ that of the free aether, the body's refractive index n is
given by

*-A./f.

PojP

But Fresnel's aether has throughout the same elasticity, within
ponderable bodies and interplanetary space, so that e = e0 and

Thus we obtain Fresnel's celebrated formula for the dragging
coefficient :

(Frsnl)

Notice that considering the excess of the aether, i.e. p- pQ per
unit volume, as a permanent part of material bodies, it can be said
simply that the aether proper is not moved at all, that it is entirely

* See the letter in question, p. 631 of reprint in Vol. II. of (Euvres completes.

38 THE THEORY OF RELATIVITY

uninfluenced by the moving bodies. Fresnel's theory is therefore
usually alluded to as the theory of a fixed aether. Implicitly, this
aether of Fresnel is supposed to be fixed relatively to the stars, or at
least to those stars which have been concerned in the aberrational
observations.

For a vacuum, or air, n = i and K = o. Thus, first of all, Fresnel's
theory is in perfect agreement with Bradley's observations. For
other media n>i and O<K<I, or the dragging is partial^ and
increases with the optical density of the medium.

By means of his dragging coefficient Fresnel treated fully the
problem of refraction in a prism, showing that it must be sensibly *
uninfluenced by the earth's motion, in agreement with Arago's obser-
vations. This problem, in fact, was the chief object of the letter
quoted.

To close his admirable letter, Fresnel gives an application of his
theory to an experiment, suggested previously, in 1766, by Boscovich,f
consisting in the observation of the phenomenon of aberration with a
telescope filled with water, — commonly called 'Airy's experiment.'
Fresnel infers from his formula for K, by simple and most elegant
reasoning, that if observations were made with such a telescope,
the aberration would be unaffected by the presence of the water.
This result was verified, for the first time, by Sir G. B. Airy in 1871,
in the observatory of Greenwich. His observations on 7 Draconis,
during 1871-1872, proved indeed that the presence of water, in place of
air, has jip sensible, i.e. no first-order (vjc) influence on the aberration.

* i.e. as far as theyfrj1/ power of v\c goes.

t R. J. Boscovich (or Boskovic), born in Ragusa 1711, died in Milan 1787.
The principle of the water-telescope was first explained by Boscovich in a letter to
Beccaria in 1766, and then fully developed in the second volume of his optical
and astronomical papers, Opera pertinentia ad opticam et astronomiam ; Basani,
1785? Vol. Ill, opusculum III. pp. 248-314. An interesting account of the work
(and life) of Boscovich is given by G. V. Schiaparelli in a manuscript, SulF
attivith del Boskovic quale astronomo in Milano, edited recently by Dr. V. Varicak
(Agram, South Slavic Acad. of Sc., 190; 1912). In connexion with the subject
of our Chap. I., the reader may also be warmly recommended to consult another
paper of Boscovich, edited by Dr. Varicak (ibidem, 190 ; 1912) : De motu absolute,
an possit a relative distingui, originally a supplement of Boscovich to Philosophiae
recentioris a Benedicto Stay versibus traditae, Libri X, ; Vol. I. p. 350 ; Rome,
J755' This paper, which is missing even in Duhem's bibliography of the subject
(Le moitvement absohi et le mouvement relatif, 1909), contains many remarkably
clear and radical ideas regarding the relativity of space, time and motion.

For both of these pamphlets I am indebted personally to Dr. Varicak.

THE BOSCOVICH-AIRY EXPERIMENT 39

Though Fresnel's own reasoning, reprinted at the end of the
present chapter (Note 4), exhausts the subject entirely, let us yet
dwell upon it a moment.

If the aether behaved in optically denser bodies as in air, i.e. if
there were no dragging at all, we should have, by the ship and shot
analogy, instead of (8),

v _sm<j>
cfn~ sin 6»'

f/n being the velocity of propagation of light in water, or in any other
medium filling the tube of the telescope. Then Airy's experiment
would have given a positive result. But he obtained precisely the
same <£ as for air. This negative result suggested to him (at least as
it is usually represented in text-books) the supposition that the
k water carries with it the aether ' with only a certain part of its
velocity, namely such that, in the above formula, we have to write
v instead of v, where

v = v/n,
so that

sin</>_ v _v

sin0 ~f/n~f1

as for air. In reality the process of compensation is not so simple as
this; but in Airy's experiment the compensation — sensibly complete —
is produced in a slightly different way. Considering a slab of water
moving perpendicularly to its axis, and neglecting second-order terms
(i.e. vi\cL= io~8), you will easily obtain*

sm4>_(v-v)c , .vn* ( }

-—=v-«>— '

where, v-v being the relative velocity of the aether and telescope,
K = vjv has been written for the dragging coefficient, as yet supposed
to be unknown. Hence, to account for Airy's negative result, i.e. to
make (9) identical with (8), we have to write (i — K)«2= i, or

as in Fresnel's formula.

* See, if necessary, for instance N. R. Campbell's Modern Electrical Theory,
Cambridge, 1907 ; pp. 293-294 (but interchange the dashes at P> C, 0, Q in his
Figure 28, which are placed the wrong way ; correct also some dashes on p. 294
and read at the bottom of the page ' presence ' instead of ' pressure.' As regards
Fizeau's experiment, amend the shocking anachronism on p. 295 : ' Fizeau tried ' —
1851 — 'to test the correctness of Airy's hypothesis' — 1871).

THE THEORY OF RELATIVITY

Thus, Airy's negative result is perfectly accounted for by Fresnel's
dragging coefficient, terms of the order of io~s being, of course,
beyond the possibility of observation.

But Fresnel's formula found also, twenty years earlier, an im-
mediate verification in Fizeau's optical interference-experiment with
flowing water.* The arrangement of the apparatus which was used
by Fizeau is seen at a glance from Fig. 5. Light from a narrow slit,
S, after reflection from a plane parallel plate of glass, A A, is rendered
parallel by a lens L and separated into two pencils by apertures in a
screen EE placed in front of the tubes T19 T2 containing running
water. The two pencils, after having traversed (towards the left
hand) the respective columns of water, are focussed, by the lens £,
upon a plane mirror Z, which interchanges their paths : the upper
pencil returns towards L by the tube T2, the lower by T^. On

FIG. 5.

emerging finally from the water, both pencils are brought, by Z, to a
focus behind the plate AA, at S' (and partly also at S). Here a
system of interference fringes is produced which can be observed
and measured in the usual way. Thus, each pencil traverses both
tubes, T^ and T2, i.e. the same thickness of flowing water, say /.
Moreover, the (originally) upper pencil is travelling always with, the
other against the current. If, therefore, v be the velocity of the
water and K the dragging coefficient, the difference in light-time for
the two pencils will be given by

cln — KV c\n + KV) '

where n is the refractive index of water. Passing from stationary to
flowing water, Fizeau observed a measurable displacement of the
interference fringes, namely with #=700 cm./sec. ; and by reversing

*H. Fizeau, Comptes rendus, Vol. XXXIII., 1851; Annales de Chimie,
Vol. LVIL, 1859.

FIZEAU'S EXPERIMENT 41

the direction of the current of water the displacement of the fringes
could be doubled. From the observed displacement it is easy to
find the difference of times A, and by equating it to the above
expression of A to find the dragging coefficient K in terms of /, ;/, ?-,
which can be measured. The result of Fizeau's experiment was that
K is a fraction, sensibly less than unity. How much less, could not
be ascertained with sufficient precision. Fizeau's experiment was
therefore repeated in a form modified in several important points by
Michelson and Morley* (1886), who found, for water (moving with
the velocity of 800 cm. per second) at 18° C, and for sodium light,

K = 0-434 ±0-02, (MM)

i.e. 'with a possible error of ±0-02.'

Now, n being, in the case in question, equal to 1-3335, Fresnel's
formula gives

K= i -4, = 0-438, (Frsnl)

a value agreeing very closely with Michelson and Morley's experi-
mental result.

Thus, Fresnel's formula, deduced from what in our days may be
deemed an assumption of naive simplicity, proved to be in admirable
conformity with experiment, like everything predicted by Fresnel in
optics^ His dragging coefficient has acquired a special importance
in recent times, and every modern theory is proud to furnish his *,
which has become, in fact, one of the first requirements demanded
from every theory of electrodynamics and optics of moving bodies
which is being proposed. 'Agreeing with Fresnel' has become
almost a synonym of 'agreeing with experience.'

Now Maxwell's and Hertz-Heaviside's equations for moving media,
(Mx) and (HH), giving, as we have just seen, *=4 and *c= i, or half
and full drag, respectively, for any medium, be it as dense as water
or glass or as rare as air, proved thereby to be in full disagreement
with Fresnel, i.e. with experiment.

The first successful attempts to smooth out this discordance of
(Mx) and (HH) from experiment, which — as has been mentioned —
manifested itself also in the case of electromagnetic experiments
properly so called, were made by H. A. Lorentz in 1892. The

* Michelson and Morley, American Journ. of 'Science, Vol. XXXI. p. 377 ; 1886.
See also A. A. Michelson's popular book, Light Waves and their Uses ; Chicago

1907 ; p- 155-

42 THE THEORY OF RELATIVITY

theory proposed in a paper published in that year,* and which led
with sufficient approximation to Fresnel's dragging coefficient, was
then simplified and extended in 1895, m a paper f which has since
become classical.

Stokes' moving aether (1845) leading to serious difficulties,!
Lorentz decided in favour of Fresnel's immovable, stationary aether,
as the all-pervading electromagnetic medium.

Thus, Lorentz's theory, presently known widely as the Electron
Theory, is, first of all, based on the assumption of a stationary,
isotropic and homogeneous aether. In calling it shortly ' stationary '
(ruhend\ Lorentz states expressly that to speak of the aether's
4 absolute rest ' would be pure nonsense, and that what he means is
only that the several parts of the aether do not move relatively to one
another (Essay , p. 4). In other words, Lorentz's aether is not
deformed, it is subjected to no strain, and does not, consequently,
execute any mechanical oscillations. And this being the case, it has,
of course, no kind of elasticity, nor inertia or density. It is thus far
less corporeal than Fresnel's aether. One fails to see what properties,
in fact, it still has left to it, besides that of being a colourless seat
(we cannot even say substratum) of the electromagnetic vectors E, M.
And although Lorentz himself continues to tell us, in 1909,! that he
1 cannot but regard the ether as endowed with a certain degree of
substantiality,' yet, for the use he ever made of the aether, he might
as well have called it an empty theatre of E, M, and their perform-
ances, or a purely geometrical system of reference, stationary with
regard to the (or at least to some) ' fixed ' stars. This aether, having
been deprived of many of its precious properties, was at any rate
already so nearly non-substantial, that the first blow it had to sustain
from modern research knocked it out of existence altogether, — as will
be seen later. Still, substantial or not, for the theory of Lorentz we
are now considering, it is something, namely its unique system of
reference. So long, therefore, as it was thought that there is such an

* H. A. Lorentz, La thtorie flectroinagnttique de Maxwell et son application
aux corps motivants ; Leiden, E. J. Brill, 1892 (also in Arch. ngerL, Vol. XXV.).

t H. A. Lorentz, Versiich einer Theorie der elect rise hen ^lnd optischen Erschein-
zingen in bewegten Korpern; Leiden, E. J. Brill, 1895. This paper will be
shortly referred to as 'Essay* (= Versuch],

% See Note 5 at the end of this chapter.

§ Lorentz, The Theory oj Electrons •, etc. , Lectures delivered in Columbia
University, 1906; Leipzig, Teubner, 1909; p. 230.

LORENTZ'S EQUATIONS 43

unique system, Lorentz's all-pervading medium could continue its
scanty existence.

For this free aether, i.e. where it is not contaminated by the
presence of ponderable matter, Lorentz assumes the exact validity of
Maxwell's equations, (4), i.e.

— = c . curl M ; -^-7 = —c. curl E ; div M = o,

Of Ct

with /) = divE = o. (As to terminology, Lorentz calls the above E
the dielectric displacement ', and M the magnetic force.}

Then, to account for the optical and, more generally, electro-
magnetic phenomena in moving ponderable matter, he has recourse
to electro-atomism, an hypothesis already employed (1882-1888) by
Giese, Schuster, Arrhenius, Elster and Geitel, and others, and later
also by Helmholtz (1893) in his famous electromagnetic theory of
dispersion, and in various writings of Sir Joseph Larmor. According
to Lorentz, matter by itself has no influence whatever on the
electromagnetic phenomena : in this respect it behaves like the free
aether. Only when and as far as matter is the seat of 'ions,' in
Lorentz's, or electrons in modern terminology,* it modifies the
electromagnetic field and its variations. In other words, Maxwell's
equations, (4), are assumed to be strictly valid not only in the free
aether, but also in all those portions of ponderable molecules in
which there is no charge, i.e. wherever p = o. And as to the question
whether ponderable matter consists entirely of electrical particles
(charges) or not, Lorentz leaves it an open question. If I may
venture an opinion, it was very wise of him not to have had
M. Abraham's ambition to construct a purely electromagnetic
" Weltbild,' as the Germans call it. (This remark will be under-
stood better later on, when we shall see that, as far as we know,
even the mass of the free electrons, such as the kathode ray- or
/3-particles, may not be of purely electromagnetic origin.) The part
played in Lorentz's theory by matter itself consists only in keeping
the electrons, or at least some of them, at or round certain places,
say, restraining them from too wide excursions. Maxwell's equations,
as written above for the free aether, are modified only where

div E = p =f o,

*' Electron ' is due to Johnstone Stoney (1891). The distinction made now
between ' ions ' and ' electrons ' does not concern us here ; besides, it is generally
known from a host of popular writings.

44 THE THEORY OF RELATIVITY

i.e. where there is, at the time being, some electric charge or
electricity, and where, moreover, the electricity is moving.* The
* modification is the slightest imaginable,' to put it in Lorentz's own
words (Electron Theory, p. 12). If p be the velocity of electricity
at a point, relatively to the aether, i.e. relatively to that system of
reference, S, in which the free-aether equations (4) are valid, then
the left-hand member of the first of these equations, or the displace-
ment current, is supplemented by the convection current, per unit
area, i.e. by pp, while the second and third equations remain
unchanged.

Thus, Lorentz's differential equations, assumed to be valid exactly
or microscopically^ throughout the whole space, are

^T + PP = c . curl M, where p = div E

I /T \

3M I

-—=-<:. curl E ; div M = o.

These have been since generally called the fundamental equations of
the electron theory. They contain, of course, the equations for the
free aether as a particular case, namely for p = o.

An important supplement to the above system of equations con-
sists in the formula for the ponderomotive force ' acting on the electrons
and producing or modifying their motion,' which, guided by obvious
analogies, Lorentz assumes to be, per unit volume,

(ii.)
or, per unit charge,

(10)

This ' force ' is supposed to be exerted by the aether on electrons
or matter containing electrons. Vice versa, as Lorentz states it
expressly, matter, whether containing electrons or not, exerts no
action at all on the aether, — since the aether has already been
supposed to undergo no deformations, etc. Of course, Lorentz's
aether is massless as well. Lorentz tells us, with emphasis, not to

*This, of course, implies the possibility of our following an individual portion
or element of charge in its motion, — a subtle point (due to circuital indeterminate-
ness, etc. ), which, however, need not detain us here.

fTo be contrasted afterwards with his macroscopic (or average) equations.

LORENTZ'S EQUATIONS 45

bring in even the notion of a ' force on the aether.' It is true — he
adds — that this is against Newton's third law (action = reaction),
' but, as far as I see, nothing compels us to elevate that proposition
to a fundamental law of unlimited validity' (Essay, p. 28).

But there is no need to keep in mind all these, and similar, re-
marks and verbal explanations, — especially as the absence of force
on the free aether is seen from (11.) at a glance, by putting /> = o.

It is perfectly sufficient to state that the basis of Lorentz's
theory is entirely contained in the above (microscopically valid)
equations (i.), (n),* all other things being obtained from these
equations by more or less pure deduction, without new hypo-
theses.!

Notice, in passing, that (i.) is not a complete system in the sense
of the word explained in Chap. I. For to trace the electromagnetic
history, not only E0, M0 for / = o and for the whole space, but also p
and p for all values of / must be given. In (i.) we have, essentially, two
vector equations of the first order for three vectors E, M, p, and the
formula (n.) does not complete the system, since, on further research,
it does not lead to an equation of the form Sp/3/ = 12 (E, M, p), + but
in the most favourable case to an integral equation extending over
a certain interval of time, generally finite, but sometimes indefinitely
prolonged. But this 'incompleteness' is no disadvantage in (i.),
(ii.), especially for the purpose of macroscopic treatment, in which
consisted Lorentz's main object of constructing these equations.

The equations assembled in (i.), which, together with the formula
for the ponderomotive force, have been received into the domain
of modern Relativity, as will be seen later, can be easily condensed
into a single quaternionic equation. First of all, put

B = M-*E (u)

(where t = \/ - i ), and call it the electromagnetic bivector. Also write,
for convenience,

l=ict. (12)

* These are also the equations of Larmor, who started from the conception of a
quasi-rigid aether and deduced the equations in question from the principle of
least action. (Aether and Matter •, Cambridge, 1900.)

tTill he comes to Michelson and Morley's famous interference experiment.

£f2 being some space-operator and E, M, p the instantaneous values of the
three vectors or vector-fields.

46 THE THEORY OF RELATIVITY

Then, the first and third, and the second and fourth of (i.) coalesce
respectively into the bivectorial equations

and

div B = - ip ;

or, in Hamilton's symbols,

SVB = - (VB) = - div B = ip.

Add up, and remember that the full quaternionic * product ' of the
Hamiltonian V and of the bivector B is

then

Next, introduce the operator

which will turn out to be of fundamental importance for our subse-
quent relativistic considerations, and the quaternion

('4)

which we may call the current-quaternion. Then the last equation
becomes

£>B=C. (i.a)

Thus, the four vectorial equations in (i.) coalesce into a single
quaternionic equation (i. a), whose form will be most convenient for
relativistic electromagnetism. It is scarcely necessary to say that
what we have done here has nothing to do with Relativity. We are
not as yet so far. (i. a) is simply a formal condensation of the
fundamental electronic equations (i.).

What we are mainly concerned with in the present chapter is
the macroscopic or average result of these equations and of the force
formula (n.). But before passing to consider Lorentz's macroscopic
equations, it will be good to dwell here a little upon the exact or

ELECTROMAGNETIC ENERGY 47

microscopic formulae (i.), (11.), and some of their immediate and
most important consequences.

First, as regards the conservation of energy, multiply the first of (i.)
by E and the third by M, both times scalarly. Then, remembering
that, by (n.), /o(Ep) = (Pp), and, by vector algebra,

(E curl M) - (M curl E) = - div VEM,
the result will be

where u = $(E? + M*) (16)

and g=rVEM. (17)

Now, (Pp) is the activity of the ponderomotive force or the
work done ' by the ether on the electrons ' per unit time, and unit
volume. Thus, by (15), the principle of conservation of energy will
be satisfied for every portion of space, however small, if u is inter-
preted as the density, and at the same time Ji as tne flux> °f electro-
magnetic energy. The possibility of adding to J3 any vector of purely
solenoidal distribution need not detain us here. Ji ls widely known
as the Poynting vector, in commemoration of the fact that this
vector and the corresponding conception of the flow of energy were
first formulated by Poynting (1884). Thus we see that the density
and the flux of electromagnetic energy, given by (16) and (17), are
in Lorentz's theory precisely as in Maxwell's and Hertz-Heaviside's
theory.

Next, as regards the pojideromotive force P, in comparison with
that of Maxwell as expressed by his electromagnetic stress, use the
first and third of the fundamental equations (i.) ; then (n.) will
become

P = pE - VE curl E - VM curl M--V — M--VE ~,

C Ot C Ct

or, introducing the Poynting vector,

P = pE - VE curl E - VM curl M - -^ ^. ( 1 8)

C~ (3£

This is the expression of Lorentz's force, equivalent, in virtue
of (i.), to the original expression (n.). Now, MaxweWs pondero-
motiTe force, per unit volume, is given by

PMX = />E - VE curl E - VM curl M. (19)

48 THE THEORY OF RELATIVITY

This is the resultant of Maxwell's well-known electromagnetic stress

f» = UR - E(En) - M(Mn), (20)

i.e. PMx= -idivf1-jdivf2-kdivf3, (21)

fn being the pressure* per unit area, on a surface element whose
unit normal is n, and f1? f2, f3 meaning the same things as fn for
n = i, j, k respectively. We do not stop here to show the equi-
valence of (19) and (21), for we shall have an opportunity to do so
later. What concerns us here is the comparison of Lorentz's with
Maxwell's ponderomotive force. From (18) and (19) we see that
the former is

P P * 3$ (22}

P~PM,-?^.

Maxwell's force on the free aether, i.e. for p = o, is, by (19) and
the system (i.), which in this case coincides with Maxwell's equations,

PMx = -VEM+-VEM,

i 3$
i.e. PMX = - ^, for p = o. (I90)

Thus, in a variable field, Maxwell's ponderomotive force on the free
aether is, generally, different from zero. The supposed existence of
such a force, which has been treated on various occasions by
Heaviside, suggested to Helmholtz the argument of his last paper,
namely an investigation of the possible motions of the free aether, f
On the other hand, Lorentz's force on the free aether is always nil,
according to his fundamental formula (n.) ; as has been already
remarked, he forbids us even to talk about a force on the aether,
since its elements are supposed once and for ever to be immovable.
According to (22) the Maxwellian force on the aether is just com-
pensated by Lorentz's supplementary term - -$$fdt. In using the

Maxwellian stress fn in his theory, Lorentz cbnsiders it, of course, as
a system of 'merely fictitious tensions' (cf. Essay, p. 29). In

* Pressure proper being counted positive, and tension proper negative.

t H. v. Helmholtz, Folgerimgen aus maxwell's Theorie tiber die Bewegungen
des reinen Aethers ; Berl. Sitzber., July 5, 1893 ; Wied. Ann., Vol. LIII. p. 135,
1894.

PONDEROMOTIVE FORCE 49

Maxwell's theory the ponderomotive actions observed in electric and
magnetic fields were physically accounted for by the tensions and
pressures of the aether. But Lorentz, in order to be consistent,
avoids considering the ' aether tensions ' as something physical, since
these would mean forces exerted by the different parts of the aether
on one another. Thus, the Maxwellian stress is to him but a con-
venient instrument for calculation.

Returning to the general case, p^o, Lorentz's ponderomotive
force (n.) may be written, by (22) and (21),

P= -idivf-jdivf^kdivfg-. (23)

It thus consists of two parts, the first of which is deducible from the
Maxwellian stress, while the second, foreign to Maxwell's theory, is
given by the negative time-rate of local change of the vector Ji/^2. It
is this second term which always compensates the Maxwellian action
on the pure aether.

Finally, to obtain Lorentz's resultant force

n= (Wr

on the whole system of electrons (T being any volume containing all
the electrons), use the expression (23), and observe that

f div iid-r = f

d<r, i= i, 2, 3,

where n is the outward unit normal of the surface <r enclosing the
region r. Also remember that

i(f1n)+j(f2n) + k(f3n) = fn3 (24)

since the Maxwellian stress is irrotational or self-conjugate. Then
the result will be

$</r, (25)

<r being supposed fixed in the aether, i.e. relatively to the framework
S in which the fundamental equations are to be valid. Formula (25)
states simply the same thing for the whole system, contained in T,
which is expressed by (23) for each of its elements. Of course, in
passing from (23) to (25), the continuity of the vector fn (or at least
of its components normal to surfaces of discontinuity, if there be any)
S.R. D

50 THE THEORY OF RELATIVITY

has been tacitly assumed throughout r* The last formula, again,
may be written :

which needs no further explanation. Now, as the mathematicians
say, let cr expand to infinity, or at least so that, £, M decreasing in
the usual way as i/r2, the surface integral may vanish. Then

nMx = o,

while

where the vector G is defined by

r T

(27)

and is called the electromagnetic momentum.

Thus Maxwell's resultant force is strictly nil, satisfying Newton's
third law (actio est par reactioni\ while Lorentz's resultant force is
generally different from zero, against the third law, — a result which
has been already stated in a slightly different form. Thus Maxwell's
theory, admitting an action on the pure aether, did, while Lorentz's
theory, denying it, does not satisfy Newton's third law. But, as was
observed by Lorentz himself, there is nothing to compel us to
universalize that law of Newtonian mechanics. At first, Poincare
tried to use this as an argument against Lorentz's theory ; f but he
soon gave it up. This was to be only one of a whole series of
sacrifices, and not the greatest one, made by modern physicists.

Similarly, the resultant moment of the ponderomotive forces,

(28)

where r is the vector drawn to any point of the field from a point O
fixed in the aether, or fixed relatively to S, may be easily put into the
form

*The treatment of possible exceptions to this assumption, as electromagnetic
surfaces of discontinuity or -waves properly so called [which exceptions seem to be
overlooked by the leading electronists, who claim for (25) general validity], need
not detain us here.

fH. Poincare, Arch. Nterland.^ Vol. V. ; 1900.

ELECTROMAGNETIC MOMENTUM 51

Thus, for the whole space, and with the usual assumption as to the
behaviour of £, M at infinity,

and

where

H =

is called the electromagnetic moment of mome?itum. Its analogy to
the ordinary, mechanical, moment of momentum

is obvious. So is also the analogy of the above G with the ordinary
momentum

2wv

and the corresponding interpretation of (26) and (29). Both G- and
H are so constructed as if the aether contained (electromagnetic)
momentum in each of its elements amounting to

(30)

per unit volume.

So much as regards the chief consequences of the fundamental
formulae (i.) and (H.).

Now for Lorentz's macroscopic equations. These are obtained
from (i.), (n.) by averaging over ' physically infinitesimal ' regions of
space. Lorentz calls a length / 'physically infinitesimal' (in dis-
tinction from what is called 'mathematically infinitesimal') if the
values of any observable magnitude obtaining in two points distant /
from one another are sensibly equal to, i.e. indiscernible from, one
another. Molecular, and, a fortiori, electronic, dimensions and
mutual distances of molecules constituting a ponderable medium, are
assumed to be small fractions of /. Let ^ be any magnitude, scalar
or vectorial. Round a point P draw a sphere of physically infinitesimal
radius ; let T be the volume of this sphere. Then

is called the ' mean value of ^ at P,1 and is denoted by ^. If ^ be
any of the magnitudes involved in the fundamental (microscopic)

52 THE THEORY OF RELATIVITY

equations, as for instance p or M, then ^ is what is macroscopically
observable.

We cannot reproduce here the details of the process of averaging
based upon the above fundamental notion,* but shall simply write
down the resulting macroscopic equations, limiting ourselves to the
case of a 'perfectly transparent (i.e. non-conducting), non-magnetic
ponderable medium, and leaving out of account dispersion. We
must, however, explain first the meaning of the symbols involved in
these equations.

Assuming that the molecules of the ponderable medium or body
contain electrons,! to which belong certain positions of equilibrium
within the individual molecules, Lorentz supposes their displacements
from these positions, q, and their velocities relative to the cor-
responding molecule,

to be infinitesimal. In other words, he neglects the squares and
products of q, q, or any of their components in presence of their
first powers. Notice that the only part played by the molecules of
ponderable matter consists here in restraining the electrons, i.e. in
keeping them near certain positions. For, as has already been
remarked, one of Lorentz's fundamental assumptions is, that matter
by itself, apart from electricity, behaves like the free aether, its
presence having no influence whatever upon the electromagnetic
field.

Let e be the charge of an electron which has experienced
the displacement q, as explained above. Then Lorentz brings
in the notion of electrical moment, not unfamiliar to older
theories, defining this vector to be, per unit volume, the average
of eq,, i.e.

^q.

Taking the sum of this and of the average of our above E, Lorentz
introduces the macroscopic vector

<£ = E + ^q (31)

*See Sections II. and IV. of Lorentz's Essay, or his article in Encykl. d.
math. Wiss., Vol. V.2, pp. 2OO et seq. ; Leipzig, 1904.

fViz. 'polarization-electrons,' and leaving out of account circling or ' mag-
netization- ' and free or 'conduction-electrons.'

LORENTZ MACROSCOPIC EQUATIONS 53

which he calls the dielectric polarization* Thus, in the free aether
<£ reduces to E, and generally (£ is what Maxwell called the
dielectric displacement.

Next, the macroscopic magnetic force is denned to be the average
of our above M, i.e. M, instead of which, however, we shall write
shortly M.

Finally, the macroscopic electric force is introduced, being denned
as the average of E', i.e. of the ponderomotive force per unit charge,
as given by the formula (10). Instead of E' we shall, again, write
more conveniently E'. Thus Lorentz's macroscopic electric force
will be

(32)

Notice that here p means the resultant velocity of an electron, i.e.
the vector sum of its velocity relatively to the molecule in question
and of the velocity of the ponderable body as a whole, say v,
'relatively to the aether,' so that p = q + v.

With these meanings of the symbols, Lorentz's macroscopic
equations for a transparent, non-magnetic, ponderable body, moving
with constant^ velocity v * through the stagnant aether,' i.e. relatively
to the framework S, are as follows (Essay, p. 76) :

(33)

-75— = c . curl M' ; div (£ = o

3M

-^— = - c . curl E' ; div M = o

of

M' = M - * VvE'

' /*± *

c

Here the system of coordinates involved in div and curl, is rigidly
attached to the ponderable body, thus sharing in its motion through
the aether. But the time / is the same as in the fundamental
equations (i.); obviously, therefore, 3/3/ is the time rate of change
(for constant values of those coordinates, />.) at a fixed point of fhe
body, not of the aether or of S.

*The above <£ is Lorentz's ^.

t Constant in space and time, that is to say for a body having a uniform purely
translational, rectilinear motion.

54 THE THEORY OF RELATIVITY

The second of (33) is an obvious expression of the (assumed)
absence of macroscopic charge, i.e. of /5 = o. In the more general
case of a sensibly charged body we should have div (& = p, where J> is
the observable density. As to K, appearing in the last of (33), it is a
linear vector operator in crystalline, and a simple scalar coefficient in
isotropic bodies, known as the ' dielectric constant ' or permittivity,
and depending in a complicated way on the distributional properties
of the electrons. The numerical value of K in an isotropic, and its
principal values, K^ K^ Kz in a crystalline body, are not constant,
of course, but vary with the period T of the incident light- or,
generally, electromagnetic oscillations. However, to avoid unneces-
sary complication, we may think here of the simple case of homo-
geneous light, of a particular kind (colour). Then K, or K^ A"2, K^,
are constants, whose numerical values are to be considered as
deduced from the observable refractive properties of the body with
regard to light of that particular kind. In case of isotropy we have
to write K=riL, if n be the corresponding index of refraction.*

Notice that (33) contains, besides the solenoidal conditions for
<£ and M, four vector equations for as many vectors,

<B, M, E', M',

the velocity of motion v being given. And since the differential
equations are of the first order with regard to /, the electromagnetic
history of the whole medium is determined by its initial state, say, by
<£0, M0 given for /=o.

It must be kept in mind that, to obtain the system of equa-
tions (33) from the fundamental ones, Lorentz has consciously
neglected not only various small terms concerning the minute
influence of electrons, but also all terms of second order in /3, or, to
put it shortly, all fit-terms, where

This is especially true of the fifth of (33), which has been obtained
from the more exact formula M' = M - VvE/<r by writing E' instead
of E, and thus [cf. (32)] omitting VvVpM/<r2, which is a /52-term.

* As to dispersion, which need not detain us here, it can be accounted for in the
well-known way by attributing to the body (or to its molecules) one or more
internal, 'natural periods,' and, to introduce these, plenty of opportunities are
offered by the hypothesis of the electronic structure of molecules and atoms.

LORENTZ MACROSCOPIC EQUATIONS 55

Let us now consider some of the most important consequences
flowing from the above system of macroscopic equations.

First of all, as the reader may easily find by himself,* they give
the right value for the dragging coefficient, viz. sensibly Fresnel's

coefficient, K= i --^. This, in fact, is a consequence of (33), when

/32-terms are neglected and when dispersion is not taken into account.
For a dispersive medium that value of the index of refraction is to be
taken which corresponds to the ' relative ' period of oscillation, T', —
a concept to be explained further on. This gives a slight correction
term, — n~ ~lTdn /3 T (Essay, p. 101), where n is the refractive index of
the medium corresponding to the 'absolute' period T, i.e. the
period of the oscillations emitted by the source, say, in Fizeau's
experiment. Thus, Lorentz's formula is

For water, at i8°C, and for sodium light, this becomes

" = 0-451, (Lor)

whereas Fresnel's value, and that obtained experimentally by
Michelson and Morley, have been 0-438 and 0-434 ± 0-02 respec-
tively. Thus Lorentz's dragging coefficient agrees with the experi-
mental value (MM) quite as well as Fresnel's, especially if the
'possible error of ±0-02' be taken into account. In a word,
Lorentz's equations give the right value of the dragging coefficient. And,
from what has been said previously, it can be argued that these
equations will also give correct results for all first order phenomena.
Next, putting v = o, we see at once that (33) become

cX£
3/

^=-<-.cur!E; divM = o ( (33o)

that is to say, identical with Maxwell's equations, for a station-
ary (non-magnetic) medium, (i), p. 24. Taking account of

* Proceeding, mutatis mutandis, similarly as in Note 3, concerning (HH).
Another, more simple, method of obtaining the dragging coefficient is to apply
Lorentz's 'theorem of corresponding states,' to be considered later.

56 THE THEORY OF RELATIVITY

magnetization-electrons, we would have, in the second and third
equation, JE instead of M, where Jft = /*M, //. being the permeability.

This is a very satisfactory result, for, as already mentioned, Maxwell's
equations for stationary media, agreeing fully with experiment, have
been able to stand even the severe criticism of the modern relativists,
who have adopted them without the slightest modification whatever.

' Stationary ' means, of course, in Lorentz's theory, fixed relatively
to the aether.

In order to exhibit the properties of his equations, (33), in the
general case of any constant v, i.e. for a material medium having any
uniform motion of rectilinear translation relative to the aether,
Lorentz transforms these equations by introducing instead of the
time t a new variable of very remarkable properties. This, the so-
called ' local time,' which was to become one of the most immediate
forerunners of Einstein's relativistic theory, deserves a rather more
extended treatment. It will occupy our attention in the next
chapter.

NOTES TO CHAPTER II.

Note 1 (to page 28). Let cr be a surface of electromagnetic discon-
tinuity of first order, for example ; that is to say, the vectors E, M being
themselves continuous across cr, let their space- and time-derivatives of
first order be different in absolute value and direction on the two sides
of the surface. Call one of its sides I, and the other 2 ; draw the normal
unit vector n from I towards 2, and denote by [a] the jump of any magni-
tude a, i.e. the difference 012-04. Then the so-called identical conditions,
to be fulfilled in any case, are

[div E] = (ne) ; [curl E] = Vne ; (a)

and the kinematical condition of compatibility, valid under the supposition
that the surface is neither being split into two or more nor dissolved, is

6 being the same vector as in (a\ characterizing the electrical discon-
tinuity, and b (an independent scalar) the velocity of propagation of cr,
counted positively along n. Both e and i) remain so far indeterminate,
in numerical value and direction. Similarly, for the magnetic discontinuity,

[divM] = (nm), [curl M] = Vnm, (^,)

WAVE OF DISCONTINUITY 57

m being a new vector and to the same scalar as above, since the electric
and magnetic discontinuities are supposed not to part from one another.
(For the deduction of the above conditions see J. Hadamard's Lemons sur
la propagation des ondes et les equations de Vhydrodynamiquc, Paris, 1903,
or, in vectorized form, my book on Vectorial Mechanics, London, Mac-
millan £ Co., 1913 ; also Annalen der Physik, Vol. XXVI., 1908, p. 751
and Vol. XXIX., 1909, p. 523.)

If 6, m are normal to <r, we have a longitudinal, and if tangential, a
transversal discontinuity.

So far everything has been independent of any electromagnetic con-
nections. Now use Maxwell's equations (4), with (4^ ; since they are
valid on both sides of cr, we have also

, etc.,
and, using (a\ (b) with their magnetic analogues,

(mn)=o ; (en)=o.

Notice that if b does not vanish, i.e. if there is propagation at all, the
second pair of equations becomes superfluous, since it then follows
identically from the first pair. Now, eliminating m from the first pair
of (c), we have

|«j

-2 e = VnVen = e - n(en),

n being a unit vector. But (en) = o ; hence

and similarly

-m=m

Thus, if e, m do not vanish, i.e. if there is at all a discontinuity,

that is to say, each element da- of the wave is propagated normally to
itself with the velocity c. Q.E.D.

Notice that the sign of to, left undetermined in (d\ due to the quadratic
result of elimination, may be defined uniquely by means of the original
pair of equations (c\ which are linear in to. In fact, multiply the first
scalarly by e (or the second by m), then

to = s (eVmn) = .j-(nVem),

where s is a positive scalar, namely c/e2. Thus, if n, e, m is a right-
handed system, like the usual i, j, k then to is positive, i.e. the sense of

58 THE THEORY OF RELATIVITY

propagation is that of n, and if n, e, m is left-handed, then the propaga-
tion is along — n. Thus, the sense of propagation coincides always
with that of the vector

Vem.

If e points upwards and m to the right, then the wave is propagated
forwards. Notice the similarity with the sense of the flux of energy, or
the Poynting vector, in relation to E, M,

Finally, notice, in passing, that by the first pair of (c\

similarly to the known characteristic, E'2 = M\ of the usual 'pure3 waves.
The above results may easily be extended to waves of discontinuity of
any order.

Note 2 (to page 30). Take as a surface element the parallelogram
constructed on two coinitial line elements a, b, composed always of the
same particles, so that, n being its positive normal,

Write, generally, R for (£ or JE. Then the induction through da- will
be given by the volume of the parallelepiped R, a, b, i.e.

(RnX<r=(RVab).

The current through dcr, say (pn)rt<T, being the rate of change of this
induction, is

(pn) da- = (Rn) da- + (RVab) + (RVab), (a)

where the dots stand for individual variation. Thus

and \Vectorial Mechanics, Chap. V., formula (75)]

b = (bV)v.

Now, i, j, k being the usual right-handed system of mutually normal
unit vectors, take a rectangular rfb-, say

and, consequently,

n = i, da-=dy.dz.
Then

CURRENT IN MOVING MEDIA 59

so that the sum of the last two terms in (a) will be

or, per unit area,

hence, substituting (£) in the first term of (a) and remembering that
(pn)=(pi)=A,

with similar expressions for P^p^ \t d& be taken normal to j or k respec-
tively. Thus the resultant current will be

p = current (R) = ^ + (vV)R-(RV)v + Rdiv v,
or

p = current (R) = -^ 4- V di v R + curl VR v, (c)

which is the required formula.

In the simplest case, considered on p. 33, in which the material medium
moves as a whole with purely translational velocity v=-z/i, we have to
take only the first term of (a\ so that in this case

Note 3 (to page 34). Take E, etc., proportional to an exponential function
of the argument

*-C*-*0,

where g is an imaginary constant, as usual. Then

and, consequently, curl = W =g Vi. Introducing this in the equations
(Mx), remembering that v = vi and omitting the common factor g^ we
obtain at once

60 THE THEORY OF RELATIVITY

but divJR=o gives in the present case (JEi) = o. Thus

and, the medium being isotropic,

Eliminate E, remembering that (Mi) = o; then the result will be

where to' would be the velocity of propagation, if the medium were
stationary in 5. Thus

and, the sense of propagation being that of V,

which is the required formula.

Note 4 (to page 39). To spare me trouble and to give the reader
a sample of Fresnel's charming manner of exposition, I quote here
simply the closing passages of his letter to Arago (loc. cit. pp. 633-636),
in which he treats in a masterly manner the water-telescope experiment,
both on the corpuscular and on the undulatory theory of light :

' Je terminerai cette lettre par une application de la meme theorie a
1'experience proposee par Boscovich, consistant a observer le phenomene
de 1'aberration avec des lunettes remplies d'eau, ou d'un autre fluide
beaucoup plus refringent que 1'air, pour s'assurer si la direction dans
laquelle on aperc,oit une etoile peut varier en raison du changement que
le liquide apporte dans la marche de la lumiere. Je remarquerai d'abord
qu'il est inutile de compliquer de 1'aberration le resultat que Ton cherche,
et qu'on peut aussi bien le determiner en visant un objet terrestre qu'une
etoile. Void, ce me semble, la maniere la plus simple et la plus commode
de faire 1'experience.'

' Ay ant fixe a la lunette meme, otaf plutot au microscope FBDE [figure
2 of Fresnel's letter], le point de mire M, situe dans le prolongement de
son axe optique CA, on dirigerait ce systeme perpendiculairement a
Pecliptique, et, apres avoir fait 1'observation dans un sens, on le retour-

BOSCOVICH'S EXPERIMENT

61

nerait bout pour bout, et 1'on ferait 1'observation en sens contraire. Si le
mouvement terrestre depla^ait 1'image du point M par rapport au fil de
1'oculaire, on la verrait de cette maniere tantot a droite et tantot k gauche
du fil.'

' Dans le systeme d'emission, il est clair, comme Wilson 1'a deja re-
marque, que le mouvement terrestre ne doit rien changer aux apparences
du phenomene. En eflfet, il resulte de ce mouvement que le rayon
partant de M doit prendre, pour passer par le centre de 1'objectif, une
direction MA' telle que 1'espace AA' soit parcouru par le globe dans le
meme intervalle de temps que la lumiere emploie a parcourir MA', ou
MA (k cause de la petitesse de la vitesse de la terre relativement a celle

CC'G g

de la lumiere). Representant par v la vitesse de la lumiere dans 1'air, et
par / celle de la terre [i.e. our c and v respectively], on a done :

MA : A A' \\v\t ou

AA' t
MA v

c'est le sinus d'incidence. i/ etant la vitesse de la lumiere dans le milieu
plus dense que contient la lunette \y' is our cjri\> le sinus de Tangle de

refraction C'A'G sera egal a -^ ; on aura done C'G=A'C'—, ; d'ou 1'on
tire la proportion

C'G : A'C : : t : -J.

Par consequent le fil C de 1'oculaire place dans 1'axe optique de la lunette
arrivera en G en meme temps que le rayon lumineux qui a passe par le
centre de 1'objectif.'

So far the corpuscular or emission theory. Again :

'La theorie des ondulations conduit au meme- resultat. Je suppose,
pour plus de simplicite, que le microscope est dans le vide, ^/et d' etant
les vitesses de la lumiere dans le vide et dans le milieu que contient la

62 THE THEORY OF RELATIVITY

lunette, on trouve pour le sinus de Tangle d'incidence AM A', - , et pour

, »/ ci

celui de Tangle de refraction C'AG, — . Ainsi, independamment du
ddplacement des ondes dans le sens du mouvement terrestre,

Mais la vitesse avec laquelle ces ondes sont entrainees par la partie
mobile du milieu dans lequel elles se propagent est egale a

\i.e. in our notation i> ( i — jj J ; done leur deplacement total Gg, pen-
dant le temps qu'elles emploient a traverser la lunette, est egal k
A'C d*-

d'
ainsi

On a done la proportion C'g : A'C' :: t : d' ; par consequent Timage du
point M arrivera en ^en meme temps que le fil du micrometre. Ainsi les
apparences du phenomene doivent toujours rester les memes quel que soit
le sens dans lequel on tourne cet instrument. Quoique cette experience
n'ait point encore ete faite, je ne doute pas qu'elle ne confirmat cette con-
sequence, que Ton deduit egalement du systeme de Temission et de celui
des ondulations.'

Note 5 (to page 42). Stokes' theory of aberration (' On the Aberration
of Light,' Phil. Mag., Vol. XXVII., 1845, P- 9> reprinted in Math, and
Phys. Papers, Vol. I. p. 134) was based on the assumption that the
aether surrounding the earth is dragged by this planet in its annual
motion, in such a way that the velocity of the aether relative to the earth
is nil near its surface, and, increasing gradually, becomes equal and
opposite to the earth's orbital velocity at very considerable distances
from our planet. It is obvious that this hypothesis led at once to a
rigorous independence of purely terrestrial optical phenomena from the
earth's annual motion. But in order to explain correctly astronomical
aberration, Stokes had to assume that the aether's motion, between the
earth and the ' fixed ' stars, is purely irrotational, which assumption could
not be reconciled with the absence of sliding over the earth's surface, so
long as the aether was regarded as incompressible. It is true that this
difficulty, as has been shown by Planck, can be overcome by giving up
the incompressibility, namely by supposing the aether to be condensed
around the earth and the celestial bodies, as if it were subjected to

STOKES' ABERRATION THEORY 63

gravitation and behaved more or less like a gas. But the condensation
around the earth, required to reduce the sliding to, say, one half per cent,
of the earth's orbital velocity, would be something like el\ i.e. corre-
sponding to a density of the aether near the earth about 60,000 times as
great as its density in celestial space. Now, it is certainly difficult to
admit that the velocity of light is not to any sensible extent altered by
this enormous condensation of the aether around the earth.

Particulars concerning the discussion of this most interesting subject
will be found in Lorentz's book on Theory of Electrons (Chap. V.), and
in his original paper on * Stokes' Theory of Aberration in the Supposition
of a Variable Density of the Aether,' Amsterdam Proceedings^ 1898-1899,
p. 443, reprinted in Abhandlungen ub. theor. Physik, Vol. I. p. 454.

CHAPTER III.

THEOREM OF CORRESPONDING STATES. SECOND ORDER
DIFFICULTIES. THE CONTRACTION HYPOTHESIS.
LORENTZ'S GENERALIZED THEORY.

LET us return to Lorentz's macroscopic equations, for a material
medium moving relatively to the aether with uniform velocity v,

-~- = c . curl M' ;

a/

div (B = o
= - c . curl E' ; div M = o

= M--VvE'

c

(L)

In the simplest case of a medium fixed in the aether, i.e. for v = o,
these, as already noticed, become identical with Maxwell's equations
for a stationary dielectric,

a*

BM

c . curl M ; div (£ = o
- c . curl E ; div M = o

(L0)

In order to exhibit the properties of the more general equations
(L), Lorentz introduces instead of the ' universal time,' as he calls /,
a new variable /', which will now be explained.

Let O' be a point fixed in the material body, chosen arbitrarily but
once and for ever as the origin of coordinates, x\ y, z', measured

LOCAL TIME 65

along axes rigidly attached to the body. From O' draw to any
individual point of the body P'(x, y', z') the vector r', so that the
three Cartesian coordinates are condensed in

Let us call the framework of reference rigidly attached to the body
the system S'. For comparison and to impress better upon your
mind the meaning of r', take also an initial point O fixed in the
aether, i.e. relatively to the system S, and draw from O to P
the vector r, or in semi-Cartesian expansion, using the same unit
vectors as above,*

If O' is taken to coincide with O at the instant
simply

r' = r - v/.

= o, we have

FIG. 6.

Remember that the equations (L) hold for t and x\ y z (not x, y, z)
as independent variables, or, more shortly, for

r', /.

This fixes the meaning of curl, div and 3/3/, as already mentioned
in Chap. II. As regards the curls and divergences, they are, of
course, the same in x'9 y', z as in .v, y, z.

*This is always possible, since the material body or medium moves relatively
to S in a purely translational manner.

S.R. E

66 THE THEORY OF RELATIVITY

Now, r' being the above vector characterising any given point P'
of the moving body or medium, the new variable /' is defined by

/' = /-^(r'v), (i)

and is called the local time at P'. Since the scalar product in the
second term vanishes for r'J_v, the local time coincides with the
' universal ' one at all points lying on the plane passing through O'
and perpendicular to the direction of motion. But at all other
places the new and the old time differ from one another, the local
time being behind the ' universal ' time in the anterior portion of the
body, and the reverse being the case in its posterior portion (Fig. 6).
In Cartesians, if v^iz^+j^ + ke^, the local time is

or if i be taken along the direction of motion, /' = / - x'vjc1.

Notice that Lorentz's local time, as just defined, has nothing
physical about it. It is merely an auxiliary mathematical quantity
to be used instead of the 'universal' time / in order to simplify
the form of equations (L). It is constructed expressly for this
purpose, and serves it excellently.

In fact, taking instead of r', / (or #', y , z\ t)

r', /'

as the new independent variables, and denoting the divergence and
curl in terms of the new variables by

div' and curl',
we obtain, for example, by (i) and by the third of equations (L),

divM = div'M + -vcurlE'
c

= div'M--divVvE',

since curlv = o, by hypothesis. But for VvE', as for any vector
normal to v, we have, obviously, div = div'. Hence, by the fifth
of (L),

div M = div' (M - - VvE') = div' M'.

CORRESPONDING STATES 67

Thus, the fourth of equations (L), divM = o, becomes, in the new
variables, div'M' = o. Similarly, the second of (L), div(£ = o, is
transformed into div' (£' = o, where (£' is a new vector defined by
the formula

<£' = <£+ 1 VvM. (2)

Using this new vector and the vector M', denned by the fifth
equation, the remaining equations (L) may be transformed, with
equal ease, to the new variables.

The result is surprisingly simple. The system of Lorentz's
equations (L) for a moving medium takes with the new variables
r', t'(x',y, z, /) the form

-^-T- = c . curl' M' ;
ut

r = - c . curl' E' : div' M' = o

-~-
of

(L')

that is to say, precisely the same form as for a stationary medium,
(L0), the only difference being that the electromagnetic vectors E,
(£, M are replaced by their dashed correspondents, as are also
the independent variables r, /.

This remarkable discovery, made by Lorentz, has played a most
important role not only in his own theory, but also in the subsequent
evolution of ideas concerning electromagnetism and optics. Un-
doubtedly, it may, to a great extent, be regarded as the germ of
modern relativistic tendencies. It will therefore be worth our while
to treat this subject at some length, and not only as an historical
episode.

The above result may be put into the form of what has been
called by Lorentz the Theorem of corresponding states :

If we have for a stationary medium or system of bodies any
solution (of Maxwell's equations L0), in which

E, <£, M

are certain functions of

x, y, s, /,

68 THE THEORY OF RELATIVITY

we will obtain a solution for the same system of bodies moving
with uniform translation-velocity v, taking for

E', <£', M'

exactly the same functions of the variables

x', y, z and t' = t-\ (vr').

In other words, and somewhat more shortly :

For each state in which E, (£, M depend in a certain way on
xi y\ z-> t m tne stationary system, there is a corresponding state in
the moving system characterised by E', (£', M' which depend in
the same way on x , y', 2', t'.

It will be useful to put here together the scattered definitions of
the dashed vectors. These are, by (32), Chap. II.,* by (2) and by
the fifth of equations (L),

-VvM

c

* VvM

- VvE'.

c

(3)

As to the coordinate systems, notice that they are in both cases
rigidly attached to the material medium or to the system of bodies
in question, x, y, z being fixed together with it in the aether, and
x't y', z sharing its motion through the aether.

The above theorem of corresponding states has, of course, like
the equations (L) themselves, the character of a first approximation
only, terms of the order of j3'2 = v2/c2 having been neglected.

The broad and easy applicability of this beautiful theorem of
Lorentz is obvious. It will be enough to quote here a few illus-
trative examples.

* Remembering that M itself is of the first order, so that

i VpM == i VvM = ^ WM,
i.e. in the adopted short notation, -VvM.

OPTICS OF MOVING SYSTEMS 69

If, in the stationary medium or system S of bodies, E, (£, M are
periodical functions of /, with period T, then, in the moving system
S', the vectors E', (£', M' are periodical functions of the local time /',
and consequently, at a point P' fixed in S', also of /, with the same
relative period T. What Lorentz calls the relative period is the
period of changes going on at a fixed point of the system S' moving
relatively to the aether, i.e. for a constant r', whereas the period
of changes taking place at a point fixed in the aether, i.e. for a
constant r, is called the absolute period. Similarly, relative rays are
distinguished from absolute rays, and so on. Thus, to luminous
vibrations in 5 of a given absolute period correspond luminous vibra-
tions in S' of the same relative period.

If, in certain regions of the stationary system, J5 = o, etc., then
also E = o, etc., in the corresponding regions of the moving system.
Thus, to darkness corresponds darkness. Also, limitations of beams
in S and S' correspond to one another. Luminous rays in S\ of
relative period T, are refracted and reflected according to the same
laws as rays of (absolute) period T in S. The same is true of the
distribution of dark and bright interference fringes, and consequently
also of the concentration of light in a focus, by mirrors or lenses,
this being a limiting case of diffraction.

But, although the lateral limitations of beams for corresponding
states are the same, corresponding wave normals in S, S' have
generally dijferent directions, this being again an immediate conse-
quence of the theorem of corresponding states. In fact, if we have
in S, say, plane waves whose normal is given by the unit vector n
and whose velocity of propagation is b, i.e. if E, (£, M are proportional
to a function of the argument

(rn) - b/,

then, in the moving system, E', etc., will be the same functions of the
argument

(r'n)-b/' = (r'n) + (r'v)-b/. (4)

Consequently, the direction of the wave normal in the moving
system will be given by that of the vector

(5)

70 THE THEORY OF RELATIVITY

Thus, unless n || v, the directions of the wave normals in S and S'
are different. To state the same thing in Cartesians, the direction-
cosines of the wave normal in the moving system will be given by
the proportions

In particular, for a vacuum or, very approximately, for air, in which
case to = t,

N' = n + ^v, (50)

or, in clumsy Cartesians,

These formulae may, after a slight transformation, be applied at
once to the case of astronomical aberration, the relative period
being here that reduced according to Doppler's law. Thus Lorentz
obtains immediately the right results for air- and water-telescope
aberration. (Cf. Essay , p. 89.)

To obtain the dragging coefficient it is enough to write the
argument (4)

Since here n' is a unit vector, the velocity of propagation in S' is

or, neglecting the term containing ft2 = (vjcf, developing the square
root and neglecting again the second and higher powers of (vn)/V,

(6)

In particular, if the propagation is in the direction of motion or
against it, as in Fizeau's experiment,

»• =•,*(*)%. '

TERRESTRIAL OPTICS 71

Thus, the velocity of propagation relative to the aether will be

»{-©>

and the value of the dragging coefficient

Here v = cjb is the refractive index of the medium, say water, corre-
sponding to the relative period which is connected with the period T
of the emitted light by the formula

second order terms being neglected. Thus, if n be the refractive
index for the period T,

whence Lorentz's formula for the dragging coefficient,

i i 3«

* = I -- 9 -- -I ^7^ J

n- n 3T

closely agreeing with experiment, as already mentioned in Chapter II.
For purely terrestrial experiments, in which not only the observer
but also every part of his apparatus and the source of light are
attached to the earth, the theorem of corresponding states leads to
the following result :

The earth's motion has no first order influence whatever on any of
such experiments.

The possibility of a second order influence remains, of course, in
this stage of the research, an open question. For, as will be re-
membered, before arriving at the macroscopic equations (L), from
which the theorem of corresponding states has been seen to follow,
/^-terms have been throughout neglected. In other words, that
beautiful theorem, developed and illustrated by a series of most
important examples in the fifth section of Lorentz's classical JEssay,
is but a first order approximation.

72 THE THEORY OF RELATIVITY

So far everything is quite satisfactory. But now, in the sixth,
and last, section of Lorentz's Essay the difficulties begin. * In
this section Lorentz investigates three problems, of which two
concern the rotation of the plane of polarization and Fizeau's
polarization experiments. But without dwelling on these, we shall
pass straight on to the third one, namely to the famous inter-
ference experiment of Michelson and Morley. This second order or
/^-experiment, originally suggested by Maxwell,! was performed by
Michelson in 1881, and six years later repeated on a larger scale
and writh a higher degree of exactness by Michelson and Morley. J
A beam of luminous rays coming from the source s, after having been
made parallel in the usual way, is divided by the semi-transparent

B

FIG. 7.

plane mirror (half-silvered plate) ab, which is inclined at an angle
of 45° to sOA, into a transmitted beam OA, and a reflected one OB,
After having been reflected by the mirrors placed at A and B (at
right angles to OA, OB, which directions are perpendicular to each
other), the two beams of light return to the central mirror ; here a
part of the first beam is reflected along OC and a part of the second

* As is explicitly stated in the title : ' Abschnitt VI. — Versuche, deren Ergeb-
nisse sich nicht ohne Weiteres erklaren lassen.'

t See Note at the end of chapter.

\ A. A. Michelson, ' The relative motion of the earth and the luminiferous
ether,' Amer. Journ. of Science, 3rd Ser. Vol. XXII., 1881. A. A. Michelson
and E. W. Morley, Sill. Journ., 2nd Ser. Vol. XXXI. , 1886; Amer. Journ. of
Science, 3rd Ser. Vol. XXXIV., 1887; Phil. Mag., 5th Ser. Vol. XXIV., 1887.
What is given above is but the usual rough scheme ; details of the actual arrange-
ment will be found in the original papers quoted and, to a certain extent, also in
Michelson's popular book on Light Waves and their Uses, where a diagram of
the actual apparatus is given (Fig. 108).

THE MICHELSOX EXPERIMENT 73

beam is transmitted towards C, thus producing with one another a
system of bright and dark interference fringes, which can be observed
through a telescope placed on the line OC. To resume it shortly,
the paths, taken relatively to the earth, of the two interfering beams
of light are :

sOAAOC and sOBBOC.

Let OA (Fig. 7) be in the direction of the motion of the earth,
and consequently also of the apparatus, source and all, with respect
to the aether of Fresnel and Lorentz, and let v be the velocity of
this motion, i.e. the resultant of the earth's orbital velocity, at the
time being, and of the velocity of the solar system with respect to
the 'fixed stars' or to those 'fixed' stars relatively to which the
aether is supposed to be at rest. (Cf. Note 2.) On this assumption ,
let us calculate the times taken by the two beams in travelling along I
their paths. Since the parts sO and OC are common to both, we
have only to consider the intervals of time, say T^ and T2, taken
to traverse

OAAO and OB BO

respectively, where the letters denote, of course, points attached to
the apparatus.

Now, as has been already said in Chapter II., in connexion with
Maxwell's equations for the 'free aether,' the velocity of light with
respect to the aether is always equal c=$. io10 cm. sec."1, quite
independently of the motion of its source. This is no novel idea
at all ; Fresnel himself considers it apparently as an obvious matter,
when he says (in an early part of his letter, already mentioned)
without any further explanations : ' car la vitesse avec laquelle se
propagent les ondes est independante du mouvement du corps dont
elles emanent.' Thus, according to both the classical and the more
recent adherents of the aether, the velocity of light relative to the
aether does not depend on the source's motion : and on the wave-theory
there is no reason why it should. Newton's corpuscular theory,
revived in a more elaborate form in the writings of the late Dr. Ritz,
need not detain us here.

Thus, the mirror A, receding from the waves on the part OA of
their journey, and the mirror O moving toward them on their return
from A to O, we have

—+— = -r,

c-v c+r) c1 -v1

74

THE THEORY OF RELATIVITY

where the index i is to remind us that OA is ' longitudinal,' i.e. along
the direction of motion. Putting vjc^P and

&>••/

(7)

we may write shortly, without yet making any use of the smallness
of /3s,

Ti = *--fOAi. (8)

To obtain T2, the time for the second beam, we could say simply,
after the manner of some authors, that the relative velocity of light,
being the vector sum of the velocity c parallel to OB and of the
velocity v of the aether with respect to the apparatus, perpendicular
to OB and directed backwards, is equal (c2 - vrf, so that

7;= 2

or

T2 = -cyOJ3t,

(9)

0 0' 0'

FIG. S.

where the index t is to remind us that OB is ' transversal ' or per-
pendicular to the direction of motion. But since this may not
seem very satisfactory, we can support it by the following, equally
frequent, reasoning which is but formally different from the above
short statement. Contemplate for a moment Fig. 8, the paper
on which it is drawn being now supposed to be stationary in the
aether, and the apparatus moving past it from left to right. Let
the centre of the inclined mirror be at O at the instant t = o, when
the light leaves it, and at O" at the instant /= T2, when the light
returns to it ; let B' be the position of B when the beam reaches
it, and let O' be the simultaneous position of O, If it be granted

THE MICHELSON EXPERIMENT 75

that the three distinct points of the aether, O, O\ O", are the
consecutive positions of exactly the same point of the inclined
mirror, that is to say, that the ray in question returns to exactly,
or sensibly, the same point of the mirror from which it started, then
OB'O" will be an isosceles* triangle, so that OB' = \cT^ and

This gives T* = 2OBl(c--vr)~^, which is identical with (9).

By (8) and (9) we get for the time-difference of the two beams,
by which the phenomenon of their interference is determined,

T^-T^^OA.-OB,}. (10)

Let us now turn round the whole apparatus through 90°, so that
OA becomes transversal, and OB longitudinal. Then we shall have,
using dashes to distinguish this case from the above one,

so that the time-difference of the two beams will become

(10')

If therefore the fixed-aether theory is true, such a rotation of
the apparatus should produce a shift in the position of the inter-
ference fringes, corresponding to the change of the time-difference
of the two beams, A = (io) - (10'), i.e.

The indices , and t, distinguishing between longitudinal and trans-
versal orientation, have been introduced here (contrary to the his-
torical order) only for the sake of subsequent discussions. To
Michelson and Morley there was no question of distinguishing be-
tween the lengths of a segment in different orientations. To put

* That the above assumption is satisfied with a sufficient degree of accuracy
may be seen from Note 3 at the end of the chapter, where the corresponding
Huygens construction is worked out.

76 THE THEORY OF RELATIVITY

ourselves into agreement with their manner of treatment we have,
therefore, to write simply

To secure these equalities Michelson and Morley mounted the
mirrors* and, in fact, the whole of the apparatus, on a heavy slab
of stone mounted on a disc of wood which floated in a tank of
mercury, so as to be able ' to rotate the apparatus without intro-
ducing strains.' In a word, they made the configuration of O,
A, etc., 'rigid,' that is to say as rigid as a stone is. On this
understanding, formula (n) may be written

A = -7(y-i).(O4 + a#). (12)

As to the mutual relation of OA, OB, they were made 'nearly
equal,' to suit the well-known requirements for producing neat
interference fringes, in each of the two orientations of the appa-
ratus. Moreover, since these lengths or distances enter in the
formula only by their sum, their equality or non-equality is of no
essential importance. We may therefore, without any more ado,
write OA = OB = L or else call the sum of these lengths 2Z. Then,
as regards the factor depending on the velocity of motion, we have,

<7)'

or, up to quantities of the second order, i.e. neglecting /34-terms, etc.,

Thus, the second-order effect to be expected on the stationary-
aether theory would be determined by the change of the time-
difference of the two beams

A=^Z. (120)

If T be the period of the light and A = r7^the wave-length, the
corresponding shift s = &/Tof the interference bands, measured as
a fractional part of the distance of two neighbouring bands, would be
given by

* = /82X' <I3)

* In the actual experiment not three but sixteen in number.

THE MICHELSOX EXPERIMENT 77

The length 2Z, which in Michelson's original apparatus was too
small, was in Michelson and Morley's experiment (1887) increased to
about 22 metres, by multiple reflection from suitably placed mirrors.
And since, for sodium light, A = 5«89.io~5 cm., the ratio 2Z/A. had
nearly the value -37 . io8. As regards /32, we should have, taking
for v simply the earth's orbital velocity, i.e. 30 kilom. per second,
f&= io~s. It is true that, at least in some of the experiments, the
rays of light, being horizontal, made a considerable angle with the
earth's orbit, but on the other hand the motion of the whole
solar system exerted a favourable influence, so as to double the
value of J3'2 (as was already mentioned). So that to put [P equal
io s is certainly not to overestimate its value considerably. Thus
the shift should be on the stationary-aether theory, in round figures,

s = 0-4 of a fringe width.

In no case, however, did the actual displacement of the fringes
exceed -02, and probably it was less than -or, i.e. less than Joih
of the expected value. The experiment was repeated in 1905 by
Morley and Miller* with considerably increased accuracy, and their
result was that, if there is any fringe-shift of the kind expected, it is
something like ^ = -0076 instead of 1-5, i.e. not greater than one
two-hundredth of the computed value, t

Thus, not nearly the expected second-order effect of the earth's
motion relatively to the aether was observed. It seems, therefore,
reasonable to say at least that, as far as we know, the above
A is nil.

In order to explain this negative result and to save, at the same
time, the stationary-aether theory, Lorentz has had recourse to a
peculiar hypothesis, constructed ad hoc^ which occurred to him
independently of Fitzgerald, who was the first to suggest it. \ It is

*E. W. Morley and D. C. Miller, Phil. Mag., Vol. VIII. p. 753, 1904;
Phil. Mag., Vol.' IX. p. 680, 1905.

fAs to various objections raised against the correctness of the interference
experiment by Sutherland, Liiroth and Kohl, and their refutation by Lodge,
Lorentz, Debye and Laue, see the ' Literaturiibersicht ' in J. Laub's report
* Uel>er die experimentellen Grundlagen des Relativitatsprinzips,' Jahrbuch der
Radioaktivitat und Elektronik, Vol. VII. p. 405, 1910.

£Cf. Lorentz's Essay, p. 122 (1895), where reference is made to a p£per
of his, dated 1892-93. As regards Fitzgerald, we read in The Ether of Space
by Sir Oliver Lodge (London, 1909, p. 65), referring to that hypothesis: 'It

78 THE THEORY OF RELATIVITY

now widely known under the name of the contraction hypothesis, and
it consists in assuming that, in Lorentz's words, 'the dimensions of
a solid body undergo slight changes, of the order /3'2, when it moves
through the ether,' namely a longitudinal contraction amounting
to ^P'2 per unit length or, more generally, both a transversal' and
a longitudinal lengthening, e and S, per unit length, such that
e-S = l/3'2. This would amount for the whole earth to about
6-5 centimetres only.

To see at once that the negative result of the Michelson experi-
ment is thus accounted for and to grasp as clearly as possible the
nature of the hypothesis, let us return to the more general formula
(n) for A, from which (12) or (i2«) followed by identifying OAj with
OAt, and similarly OB\ with OBt. Now, to simplify matters,
assume OBi = OAl and OBt = OAt (which, as we saw, is of no
essential importance), but on the other hand distinguish between
OAl and OAt. Then formula (n), valid by the fixed-aether theory,
will become

OAt}- (14)

and since A = o, by experience, we have to write, in order to respect
both that theory and experience,

or, up to quantities of the second order,

which is the Fitzgerald-Lorentz hypothesis.

Notice that it would be a perfectly idle thing to quarrel whether
OAt is shortened, while OAt remains unchanged, by the earth's
motion through the aether, or whether OAt alone is lengthened, or,
finally, whether both are changed in suitable proportions. The only
thing we are required by the aether theory and by experiment to do
is to consider the ratio of the lengths of one and the same ' material '

was first suggested by tfie late Professor G. F. Fitzgerald, of Trinity College,
Dublin, while sitting in my study at Liverpool and discussing the matter with
me. The suggestion bore the impress of truth from the first.' Happy are those
who are gifted with that immediate feeling for 'truth.'

THE CONTRACTION HYPOTHESIS 79

segment OA, or shortly Z, in those two orientations as being equal
to i - J/3-, or, more rigorously,

Z,:Z,Wi-/?2. (15)

This implies that for /? = o, i.e. if the earth stopped moving through
the aether, or nearly so, we should have Ll = Lt, say, both equal
to Z0. But it cannot inform us as to the ratio which either length
bears to Z0, when the earth is moving through that medium ; more-
over, such considerations are, thus far, physically meaningless.

At any rate, Lorentz soon decided in favour of a purely longitudinal
contraction, which amounts to writing

Z, = Z0 and Z, = y° = Z0s/f^ (15*)

In doing so he based himself on certain results obtained from
the fundamental (microscopic) equations in an early part of his
classical Essay, to be mentioned presently. That this, in fact, was
his choice we see explicitly from the shape attributed by him
to moving electrons. While Abraham's electron is and remains
always a sphere, being rigid in the classical sense of the word,
Lorentz's electron is a sphere of radius ^?, say, when at rest, and
becomes- flattened longitudinally, when in uniform motion, to a
rotational ellipsoid of semiaxes

•. -^, ./?, ./?.

y

Such an electron, of, homogeneous surface- or volume-charge, is now
generally known as the Lorentz electron. The history of its rivalry
with the rigid one, and of its rather victorious issue from the contest,
need not detain us here. It is, besides, sufficiently well known.

Lorentz's attitude towards the contraction hypothesis may be seen
best from his own words, written in 1909 (Electron Theory, p. 196) :

'The hypothesis certainly looks rather startling at first sight, but we
can scarcely escape from it, so long as we persist in regarding the ether
as immovable. We may, I think, even go so far as to say that, on this
assumption, Michelson's experiment proves the changes of dimension in
question, and that the conclusion is no less legitimate than the inferences
concerning the dilatation by heat or the changes of the refractive index
that have been drawn in many other cases from the observed positions
of interference bands.'

80 THE THEORY OF RELATIVITY

The obvious criticism of the above comparison may be left to
the reader.

As regards the justification of the contraction hypothesis which to
an unprepared mind certainly does 'look rather startling,' Lorentz
observes in his original Essay of 1895 (P- I24-) tnat we are ^e(^
precisely to the change of dimensions defined by (15^), if, dis-
regarding the molecular motion, we assume that the attractive and
repulsive forces acting on any molecule of a solid body which ' is
left to itself are in mutual equilibrium, and if we apply to these
molecular forces the same law which, by the fundamental equations,
holds for electrostatic actions. It is true, as Lorentz himself con-
fesses, that ' there is, of course, no reason ' for making the second of
these assumptions. But those who entertain the hope of constructing
an electromagnetic theory of matter will easily adhere to it. To
obtain the law in question return to the fundamental electronic
equations (i.), Chap. II., and introduce the so-called vector potential
A and the scalar potential <£, satisfying the differential equations

(i6)

) c }

and subject to the condition

(17)
Then all of the equations (i.) will be satisfied by

(18)

= curlA,

so that every electromagnetic problem is reduced to finding the
potentials according to (16) and (17). Suppose, now, that a material
body moves as a whole, relatively to the aether or to the system S,
with uniform translational velocity v, and that all the electrons it
carries are at rest with respect to the body. Then the above p will
have throughout the constant value v, so that, by (16),

A = v</>. (19)

THE CONTRACTION HYPOTHESIS 81

Thus everything is made to depend on </> alone. Take the x-axis
in »S along the direction of motion, so that v = £i, A = !/?<£, and
suppose that the electromagnetic field is invariable with respect to
the material body. This assumption will be satisfied if 4> is supposed
to depend only on the coordinates attached to the body,

Thus we shall have

3 333

and the equation for <£ will become

i 32<£ B2^ B2^
?^ + S?+3F--*

while the condition (17) will be satisfied identically. Here

?-'= (i -/**),

as above. Again, by (18),

whence the ponderomotive force per unit charge, or Lorentz's
electric force, E + /8V1M, (10), Chap. II., which we shall now denote
by Jf (since the dashed E would be misleading),

(.0

where V«iB/d£+j3/di|+kB/9f*i3/aar+... is the Hamiltonian
(here acting as the slope), taken with respect to the aether or,
which in our case is the same thing, with respect to the material
body. Thus, the electric force is derived from a scalar potential
</>/y2, precisely as in ordinary electrostatics. By the way, <j>/y2 is
called the convection potential. Notice that it is Jf, the electric
force, and not the 'dielectric displacement' E, that has a scalar
potential.

S.R. F

82 THE THEORY OF RELATIVITY

Now, supposing always fi'2 < i and consequently y real, write

x = y£, y =i~i, z' = tt (22)

and denote the corresponding Hamiltonian, i3/9#' + etc., by V.
Then (20) will become

V2<j>= -p. (23)

To adopt for the moment Lorentz's notation, call the moving
material body or system of bodies the system Slt and compare it
with a system S? which is fixed in the aether and which is obtained
from Sl by stretching all its constituent bodies, together with the
electrons, longitudinally in the ratio y:i, so that to any point
£, ry, f of S1 corresponds the point ,T', y, z of S^ and so that
corresponding volume-elements, dr and dT=ydr, contain equal
charges. Then, p and p being the densities of electric charge at

corresponding points,

, i

and, by (23),

If then <£' be the scalar, electrostatic, potential in S.2, so that

V'2<£'= -p',
we shall have

~7 '
and consequently, instead of (21), using (22),

y .v y

But the electric force in the stationary system S2 is

Therefore, using the indices \ and t to denote the longitudinal
and the transversal components of the electric forces,

<$i = £i'; £t = - (#/ = A'N/T -/32, (24)

THE CONTRACTION HYPOTHESIS 83

and since charges of corresponding elements are equal, exactly the
same relations will hold between the ponderomotive forces acting
on each electron in the moving system St and on the corresponding
electron in the stationary system S2.

This is the ' law ' alluded to. Now, suppose that it is obeyed by
the molecular forces keeping together the parts of a moving solid
which, disregarding its interior molecular and electronic motions, is
to be taken for the system Sl. Then, if the molecular forces
balance each other in the corresponding stationary body S2, they
will do so in the moving body Sl. But, by (22), S1 is the body
,& contracted longitudinally with preservation of its transversal
dimensions, exactly as in (150), and the motion would produce
this flattening 'by itself.' Whence Lorentz's justification of the
contraction hypothesis.

Thus, the longitudinal contraction, though at first manifestly
invented ad hoc, to account for the negative result of the Michelson
experiment, found a kind of legitimate support by being brought
into connexion with the fundamental assumptions of the electron
theory. But the cure of the disease has not been radical. In
fact, the idea naturally suggested itself, that the Lorentz-Fitzgerald
contraction, like an ordinary strain, might give rise to double
refraction, of the order /3'2, in solids or liquids, a property which
should be directionally connected with the earth's motion round
the sun. But here again the result of experiments has been
sensibly negative. Lord Rayleigh's* experiments (1902) with liquids
(water and carbon disulphide) as well as those with solids, with
glass plates piled together, have given no trace of an effect of
the expected rkind. At least, if there was any effect on turning
round the apparatus, it was less than jj^th of that sought for.
Rayleigh's experiment was then repeated (1904) by Brace t with
considerably increased accuracy, and the result has again been
negative : the relative retardation of the rays due to the supposed
double refraction should be of the order io~8, whereas, if existent
at all, it was certainly less than 5 . io~n, in the case of glass, and
even less than 7 . io~13, in the case of water.

To account for these obstinately negative results, and with a view
to settle the matter once and for ever, Lorentz undertook what he

*Lord Rayleigh, Phil. Mag,, Vol. IV. p. 678, 1902.

tD. B. Brace, Phil. Mag., Vol. VII. p. 317, 1904; Boltzmann- Festschrift,
p. 576, 1907-

84 THE THEORY OF RELATIVITY

thought a radical discussion of the whole subject, that is to say,
of the electromagnetic phenomena in a uniformly moving system,
not as hitherto for small values of v, but for any velocity of transla-
tion smaller than that of light, i.e. for any /3< i. Lorentz's ideas,
laid down in a paper published in 1904,* are fully developed in his
Columbia University Lectures, already quoted (p. 196 et seq.). His
aim was now to reduce, 'at least as far as possible,' the electro-
magnetic equations for a moving system to the form of those that
hold for a system at rest — always, of course, relatively to the aether —
without neglecting either /3'2- or, in fact, terms of any order whatever.
It will be remembered that even in his first approximation,
i.e. when neglecting /32-terms, Lorentz employed the ' local time '
/' = / - (vr)/<:2, or, measuring x along the line of motion,

f-'-ji*.* («)

Then the necessity of accounting for the negative result of Michelson's
interference experiment brought him to the contraction hypothesis,
according to which the longitudinal dimensions of the moving
system are reduced in the ratio i : y"1, where y = ( i - /32)~% while
the transversal ones remain unchanged. This contraction corre-
sponds to /= const, and consequently may easily be shown to be
equivalent to transforming x, y, z, the coordinates of a point with
respect to axes fixed in the aether, or the 'absolute' coordinates,
into

x' = y(x-vt), y'=y, z' = z. (b)

It is true that the transformation (a) was as yet purely formal,
and that the contraction, or (/£), was introduced by Lorentz first
ad hoc, but afterwards to be justified. But at anyrate, having
already (a) and (£), Lorentz has been naturally led to investigate in
a general way the consequences of introducing, instead of x, y, z, t,

* H. A. Lorentz, ' Electromagnetic phenomena in a system moving with any
velocity smaller than that of light,' Proc. Amsterdam Acad., Vol. VI. p. 809;
1904.

t Here, according to the original definition of ' local time,' p. 66, we should
have rigorously (instead of the coordinate x, measured in the fixed framework)

x - vt, so that /' = ( I + /32) t - -g*. But, since at that stage /32-terms were neglected,

we could write simply x instead of x-vt. The symbols x', etc., in what follows
are not to be confounded with the x', etc. , of page 66.

LORENTZ GENERALIZED THEORY 85

new independent variables, called by him the effective coordinates
and the effective time,

//==, . . , , . (25)

where y is as above and A. is a numerical coefficient of which Lorentz,
provisionally, assumes only that it is a function of v alone, whose
value equals i for v = o and differs from i by an amount of the
order fi'2 for small values of the ratio f$ = vjc* Introducing the
new variables (25) into the fundamental electronic equations, (i.),
Chap. II., and defining new vectors E', M',

!

and also, instead of the relative velocity p - v of an electric particle,
the vector

i.e. with the above choice of axes, simply

P' = 7{i?(A -»)+JA + kA}» (27)

and, instead of the density p,

P=y*.~*P, (28)

Lorentz obtained again the equations (i.) with dashes,

BE'/d/' + p'p = c . curl' M', etc.,
but with the difference that divE = /> was replaced by

, (29)

* Columbia University Lectures, p. 196. The above v, 7, X stand for Lorentz's
w, k, I respectively. A transformation equivalent to (25) was previously applied

by Voigt, as early as 1887, to equations of the form -^ ^-^-V2=o; ' Ueber das

Doppler'sche Princip, Gottinger Nachrichten, 1887, p. 41. Lorentz himself
states (loc. cit., p. 198 ; 1909) that Voigt' s paper had escaped his notice all these
years, and adds: *The idea of the transformation' (25) 'might therefore have
been borrowed from Voigt, and the proof that it does not alter the form of the
equations for \hejree ether is contained in his paper.'

86 THE THEORY OF RELATIVITY

not by div'E' = /o'. Thus, the fundamental equations for the free
aether (p = p = o) turned out to be rigorously invariant with respect
to the transformation (25), which, especially for A.= i, has since been
universally called the Lorentz transformation. The same invariance
holds also in the general case, that is to say, in the presence of
electric charges, but for the slight deviation given by (29).

Using this result, Lorentz generalized his Theorem of corresponding
states for any velocity v smaller than c, and succeeded in showing
that the theorem thus extended not only accounts for the con-
traction required by the result of the Michelson experiment, but that
it explains, among other things, why Lord Rayleigh and Brace
failed to detect a double refraction due to the earth's orbital
motion. A discussion of the formulae for the longitudinal and
transversal masses of an electron, which need not detain us here,*
led Lorentz to attribute to the coefficient X (his /) the value i,
whereby the transformation formulae (25) and (26) were reduced to

= y(x- vt\ / =y, z = z,

(3°)
"-**

and

\

, M3' = y(M3- /3E2).)

With this specialization, Lorentz's modified theory, which in its
essence was built up in 1904, satisfied the requirements of self-
consistency and accounted for the negative results of all, second as
well as first order, terrestrial experiments intended to show our
planet's motion through the aether. In other words, by modifying
and gradually extending his original theory, Lorentz obtained the
desired physical equivalence of the ' moving ' system S', with its
effective coordinates and time x\ y', z', /', and of a corresponding
* stationary ' system with its absolute coordinates and time x, y, z, /.

But still one of the two systems S, S', namely S, was privileged,
being regarded by Lorentz as 'fixed in the aether.' Their equival-
ence, as indicated persistently by such numerous experiments, was
not placed as the basis of the theory, but followed as the result of
long, laborious, and rather artificial constructions, intended to com-

*See Columbia University Lectures, pp. 211-212.

LORENTZ GENERALIZED THEORY 87

pensate gradually the pretended play of the ' aether.' For, to repeat,
Lorentz continued to assume this hypothetical medium of his classical
Essay in his extended theory, dated 1904, and adheres to it even now,
if we may judge from the last sentences of his American Lectures
(p. 230). Not only is the aether for Lorentz a unique framework
of reference, but he * cannot but regard it as endowed with a certain
degree of substantiality.' According to this standpoint, then, there
certainly is such a thing as the aether, though every physical effect
of the motion of ordinary, ponderable matter through it, being
compensated by more or less intricate processes, remains undis-
coverable for ever.

As regards the above transformation of Lorentz, we may further
notice here that Poincare made, in 1906, an extensive use of
its more general form (25) \Rend. del Circolo mat. di Palermo >,
Vol. XXI. p. 129] for the treatment of the dynamics of the electron
and also of universal gravitation. Some of Poincare's results con-
tinue even now to be of considerable interest.

In the meantime, 1905, Einstein published his paper on 'the
electrodynamics of moving bodies,'* which has since become
classical, in which, aiming at a perfect reciprocity or equivalence
of the above pair of systems, S, S', and denying any claims for
primacy to either, he has investigated the whole problem from the
bottom. Asking himself questions of such a fundamental nature,
as what is to be understood by 'simultaneous' events in a pair of
distant places, and dismissing altogether the idea of an aether,
and in fact of any unique framework of reference, he has succeeded
in giving a plausible support to, and at the same time a striking
interpretation of, Lorentz's transformation formulae and the results
of Lorentz's extended theory. Einstein's fundamental ideas on
physical time and space, opening the way to modern Relativity,
will occupy our attention in the next chapter.

*A. Einstein, AnnaL der Physik, Vol. XVII. p. 891 ; 1905.

88 THE THEORY OF RELATIVITY

NOTES TO CHAPTER III.

Note 1 (to page 72). It seems desirable to quote here after Lorentz
(Abhandlungen iiber theor. Physik, Vol. I. p. 386, footnote) a passage
from Maxwell's letter ' On a possible mode of detecting a motion of the
solar system through the luminiferous ether,' published after his death in
Proc. Roy. Soc., Vol. XXX. (1879-1880), p. 108 :

' In the terrestrial methods of determining the velocity of light, the
light comes back along the same path again, so that the velocity of the
earth with respect to the ether would alter the time of the double passage
by a quantity depending on the square of the ratio of the earth's velocity
to that of light, and this is quite too small to be observed.'

Note 2 (to page 73). Usually, at least in all text-books, it is simply
said: 'Suppose that the aether remains at rest, and let ?y = the velocity
of the apparatus, i.e. of the earth in its orbit.' For this to be correct, the
aether would have to be at rest with respect to our sun. But when astrono-
mical aberration is in question, we are told that the aether is stationary
with respect to the 'fixed stars,' say, with respect to the constellation ot
Hercules, which, I hope, is 'fixed' enough. Now, as has incidentally
been mentioned (p. 17), the sun or the whole solar system has a uniform
velocity of something like 25 kilometres per second towards that con-
stellation, which, being nearly equal in absolute value to the earth's orbital
velocity (30 klm. per sec.), certainly cannot be neglected. Thus, the
velocity (y) of Michelson's interferometer with respect to the aether would
oscillate to and fro, in half-year intervals, between considerably distinct
maximum- and minimum-values. According to Lorentz (' De Pinfluence
du mouvement de la terre sur les phenomenes lumineux,' 1887, reprinted
in Abhandlungen^ Vol. I. ; see p. 388) the resultant of the earth's orbital
and the solar system's velocity had at the time when Michelson was
performing his experiment both a direction and an absolute value ' very
favorable ' to the effect sought for, even so much as to double the displace-
ment of the fringes expected. I am not aware whether or no the defenders
and the adversaries of the aether have discussed this circumstance with
sufficient care. But at any rate it seemed worth noticing here. Of
course, it is for the adherents of the aether (and not those of empty
space) to tell us explicitly with respect to what celestial bodies, the sun,
or Hercules or other groups of stars, the aether is to be stationary, if it
be granted that the parts of that medium do not move relatively to each
other. For these stars certainly move relatively to one another.

I cannot help remarking here that it is repugnant to me to think of an
omnipresent rigid aether being once and for ever at rest relatively rather
to one star than to another. For, this medium, unlike Stokes's aether,
being non-deformable and not acted on by any forces whatever, none of
the celestial bodies, be it ever so conspicuous in bulk or mass, can
claim for itself this primacy of holding fast the aether. The bare idea

REFLECTION FROM MOVING MIRROR

89

of action exerted upon the aether by material bodies being dismissed at
the outset, there is nothing which could confer this distinctive privilege
upon any one of them. But, then, I am quite aware that what 'is re-
pugnant to think of may not necessarily be wrong altogether. There are
other reasons to be urged against the aether.

Note 3 (to page 75). Let a plane wave <r (Fig. 9) proceed towards
the inclined mirror (half-silvered plate) Oa in the direction of its motion,
i.e. from left to right. Let sO, sma represent the incident wave normals,
limiting a part of the beam of breadth Om = b, and let CLY'be the normal
to the mirror, so that B=sOX is the angle of incidence. Let the wave
reach the centre O of the mirror at the instant /=o. Let Ol and a± be
the positions of the points O and a of the mirror (both taken in the plane

of the figure) at a later instant /=T, when the wave of disturbance reaches
«!, so that

aal = OOl = vr.

Draw round O a circle with the radius

then the tangent to this circle, drawn from alt will represent the re-
flected wave, and ON will be the reflected wave normal. To obtain
the angle of reflection, & = XON, consider the triangles ONa^ and Omalt
having the side Oa^ in common and right angles at m and at N. Since,
moreover, their sides CWand avm are equal to one another, alN=Om = b,
so that the breadth of the beam remains unchanged by reflection, as for
a stationary mirror, and

where f=<«C?a1. But ^=7r/2-€ + f. Thus, the angle of reflection &
and the angle of incidence 6 are connected by the relation

(A)

90 THE THEORY OF RELATIVITY

where the angle { is determined by the given properties of the parallelo-
gram OaalOl. Writing

we have at once

Oa^= (2/r)2 + /2 + 2VTl . sin 0
and

whence

cos20=i /2 2* jnfl
sinaf (vrf vr S1]

But -t/T = 7//sin Q\(c-v\ or ljvr= /J--^--/}* so that the required formula
for C is ' Sm

(A) and (B) contain the rigorous solution of the problem, based, of
course, on the assumption of a stationary aether.

In Michelson and Morley's experiment, as treated above (Fig. 8),
20=90°, so that (B) becomes

To connect Fig. 9 with Fig. 8, notice that, according to (A), the
angle BOB' should be equal to 2^. The approximate treatment given
in connexion with Fig. 8 (p. 74) amounts to writing

*\*(BOff} = v\c=$. (c)

Now, developing (BX) and remembering that ft is a small fraction, we
have, up to quantities of the second order,

or, neglecting the third and higher powers of the small angle f,

But the term J/32 appearing in this formula for the angle would give
in the final formula for T2 only terms of the order of )83 and /34. Thus,
aiming at results which are correct only up to quantities of the second
order, we may write the last formula

mi (a{)- A

in agreement with (c). Our Huygens-construction shows then that
the treatment adopted on page 74 is sufficiently correct for the purpose
in question.

That treatment, which is given in all text-books (including also such
valuable modern works as Laue's Relativitcitsprinzip, 1913) without
any further remark, would be rigorously correct if O were, say, a point

REFLECTION FROM MOVING MIRROR 91

source of (spherical) waves spreading out in all directions, and not, as
it actually is, one of the points of a mirror at which reflection of plane
waves is taking place.

A different way of treating rigorously the above question will be found
in Lorentz's paper entitled ' De Pinfluence du mouvement de la terre
sur les phenomenes lumineux,' Arch, neerL, Vol. XXI. (1887), pp. 169-172
(reprinted in Abhandlungen iiber theor, Physik, Vol. I. pp. 389-392) and
partly also in his Columbia University Lectures, p. 194.

The discussion of our general formulae (A), (B) connecting the angle
of reflection with that of incidence, for large values of /?, may be left
to the reader as a curious exercise.

CHAPTER IV.

EINSTEIN'S DEFINITION OF SIMULTANEITY. THE PRIN-
CIPLES OF RELATIVITY AND OF CONSTANT LIGHT-
VELOCITY. THE LORENTZ TRANSFORMATION.

WE are now sufficiently prepared to grasp the meaning of Einstein's
ideas* and to appreciate their relation to the work of his prede-
cessors, especially of Lorentz.

In Chapter I. we have seen how it is possible to define the time
as a physically measurable quantity fulfilling certain reasonable
and fairly general requirements. Practically, it was the variable /
measured by the rotating earth as time-keeper or what, with a
slight correction connected with tidal friction, has been called the
'kinetic time.' It has certainly not escaped the reader's notice
that the requirements on which that choice was based had nothing
absolute or necessary about them, being merely recommended by
their simplicity and convenience. But this circumstance need not
detain us here any further. Suppose we have secured a clock
indicating, with a sufficient degree of precision, the kinetic time /.
Suppose we keep that clock at a certain place «, relatively to a given
space-framework of reference, say in a certain physical laboratory or
astronomical observatory. Thus far we have tacitly assumed that
the time /, measured by such a chronometer, is universal, if I may
say so, i.e. that it is valid for all points of space, for all parts of
any system, be it near to our clock or very far from it, be it at rest
or moving with respect to it. It is very likely that nobody has ever

* As laid down in his paper of 1905, already quoted, and then (1907) developed
by him more fully in a paper, ' Ueber das Relativitatsprinzip und die aus demselben
gezogenen Folgerungen,' Jahrbitch der Radioaktivitiit und Elektronik, Vol. IV.
p. 411. In what follows we shall refer principally to the former of these papers
by quoting simply the original numbers of its pages.

DEFINITION OF SIMULTANEITY 93

asserted explicitly this universality and uniqueness of time, but
everybody has certainly given to it his tacit consent, and would
willingly endorse it if asked to do so. As far as we know, the first
to question this pretended universality of time was Einstein.

Our clock, placed at a, indicates the time /, i.e. marks different
time-instants and measures the intervals between them, to begin
with, only at the place a, or nearly so. It is, to give it a short
name, the time ta. Suppose that some well-marked instant is
chosen as the initial instant, /a = o. Then, if any event is happening
at a or near «, we give to it that date or, as it wrere, label it
with that number ta which is simultaneously shown by the index of
our clock. We are exempted from defining what ' simultaneous ' (as
well as ' earlier ' or ' later ') means when applied to a pair of events
occurring at the same place or near that place, as the passage of
the index through a given division of the dial of our clock and the
production of an electric spark closely to it* But we do not
know, beforehand, what we are to understand by saying that of
two events occurring at places «, b distant from one another the
first occurs earlier or later than the second, or that both are
simultaneous. The meaning of these words has to be defined.
If the labelling of all possible kinds of events, occurring at distant
points, fixed or moving relatively to one another, is to be of any
use at all, wTe must establish the rules according to which we are
going to label them with the /-numbers. And first of all we have
to decide which of these events have to receive the same labels,
i.e. we have to define simultaneity at distant points.

This notion is to be defined in terms of simultaneity at the same
place, which alone is assumed to be known to us, and of some other
things or processes which are actually realizable. In other words,
distant simultaneity has to be reduced to local simultaneity by some
physical process. Abstractly speaking, the choice of such a process
is arbitrary, in very wide limits at least; but practically the choice
will be reduced to siich processes as are of possibly universal
occurrence, and wrhich are independent of the capricious peculiarities
of different sorts of matter. Einstein has chosen for this purpose
the propagation of light in vacuo. Gravitation being, chiefly due

* We need not stop here to consider such apparatus as Siemens' ' spark-
chronometer,' in which the visible marks corresponding to pairs of events are
brought very close to one another, and which enable the modern physicist to fix
with a high degree of precision their time-relations.

94 THE THEORY OF RELATIVITY

to its alleged instantaneous action, out of question, this has been,
in fact, the only possible choice. Moreover, it was not unprecedented
in the history of physics and astronomy, and it suggested itself most
obviously because the recent difficulties met with lay in the optical
and, more generally, electromagnetic departments of physics.

To an unbiassed mind the question may present itself : Why label
everything in the world with Anumbers at all? Such a question
is not altogether unreasonable, and it may deserve some careful
attention. But once we decide to attach a time-label to every event,
we are forced to reduce in some kind of way distant simultaneity to
local simultaneity (for pairs of points at rest or moving relatively to
one another), and not to delude ourselves with thinking that we
know what 'universal simultaneity' means, or that it is, in fact, a
self-consistent notion. To have initiated a critical analysis of the
concept of simultaneity at all is certainly a great merit of Einstein's.

But let us leave aside these generalities and pass to the definition
in question. We shall have to consider in the first place the
simpler case of distant points a, />, etc., in relative rest, and then
the somewhat intricate case of distant points belonging to systems
which are uniformly moving with respect to one another.

Let a, b) etc., be points or 'places' fixed relatively to one another
and with respect to a certain space-framework or system S, say, the
system of the fixed stars.* Suppose we succeeded in manufacturing
at the place a a number of equal clocks, each measuring the same,
say the * kinetic,' time / and set equally or synchronously, and that
retaining one of them at a we sent the others to b, etc., together
with an equal number of observers who are to remain at those
distant places with their clocks for ever. Then, to begin with, we
should have as many ' times ' as there are places in consideration,
4) 4) etc., valid, respectively, for the places a, b, etc., and for their
nearest neighbourhoods. For, though all of these clocks were
manufactured equally at a, we do not know whether they continue
to be * equal ' or permanently synchronous, when one of them is

*In his paper (p. 892) Einstein begins with taking, for the purpose of his
definition of simultaneity, that 'system of coordinates in which Newton's
mechanical equations are valid.' But it seems advisable not to appeal at the
outset, and in connexion with such a fundamental definition, to Newtonian
mechanics, especially as it requires, according to the relativistic view itself, some
essential, though numerically slight, modifications. On the other hand, the
physical specification of what has been called above the system S will appear
presently without recourse to any theory of mechanics.

SYNCHRONOUS CLOCKS 95

still kept at a, while the others are sent far away, to the places b,
etc. More than this, we do not know what their being synchronous
or not, when far apart, means. We have yet to fix how we are going
to test it. To invoke the preservation of rate of clocks of 'good
make ' in spite of their being carried to distant places, on the title
of the high precision of their mechanisms, would not help us out
of the difficulty. For, supposing we also decided to assert such
infallible and rigorous permanence, at different places within S,
of the mechanical laws, necessarily involved, still we should have
to verify whether the accessorial conditions of validity of those
laws (and practically there would be a host of such conditions) are
fulfilled at and round each place in question. To avoid this verifi-
cation, which soon would prove to be a difficult task, we must have
some means of testing in a direct manner the synchronism of our
distant clocks and, more generally, of correlating with one another
the times /«, 4, etc., without being obliged to enter upon the
properties and structure of the corresponding clock mechanisms.*

Now, the kind of test adopted by Einstein, and constituting at
the same time the essence of his definition of distant simultaneity,
is as follows.

Let an observer stationed at a send a flash of light at the instant
/„ (as indicated by the «-clock) towards b, where it arrives at the
instant 4 (according to the ^-clock). Let another observer send it
back from b without any delay, or let the flash be automatically
reflected at b, towards a, where it returns at the instant faf. Then
the ^-clock is said, by definition, to be synchronous with the
tf-clock, if

fa -4 = 4 -/«. (i)

This amounts to requiring, per definitionem, that 'the time*
employed by light to pass from a to b should be equal to 'the
time' employed to return from b to a. Instead of (i) we may
write, equivalently,

Thus, the instant of arrival at b is expressed by the arithmetic mean
of the a-times of departure and return of the light-signal. Such

*\Ve may notice in this connexion that Einstein's specification (p. 893):
1 eine Uhr [at ti\ von genau derselben Beschaffenheit wie die in A [a] befindliche '
is unnecessary and, to a certain extent, misleading.

96 THE THEORY OF RELATIVITY

being the connexion of the a-time and of the Mime, the clock
placed at b is said to be synchronous with that placed at a.

This definition of synchronism is supposed to be self-consistent,
for any number of clocks placed at different points of the system S,
say, besides a and b, at c, d, £, etc. To secure this consistency,
Einstein makes, explicitly, the following two assumptions :

1. If the clock at b is synchronous with that at a, then also the
clock at a is synchronous with that at b. In other words : clock-
synchronism is reciprocal, for any pair of places taken in S.

2. If two clocks, placed at a and b, are synchronous with a third
clock, placed at c, they are also synchronous with one another.
Or, more shortly, clock-synchronism is transitive throughout the
system S.

This is the way that Einstein himself puts the matter. But it
may easily be shown that the first of his assumptions will be
fulfilled if we require that 'the time' employed by the light-signal
to pass from a to b is always the same. In fact, let us denote the
tf-time, taken generally, by a instead of 4, and similarly, let us write
b instead of the general variable 4, and let us use the suffixes
d> a> r to denote the instants of departure, arrival and return. Then,
if the ^-clock is synchronous with the «-clock, we have, by definition,
or

for the ' return ' at a may be equally well considered as an arrival
at that place. Now, if at the instant aa the flash be sent again
towards b, where it arrives at the instant br, we have, by our above
requirement,

ba-ad = br-aa,

and, by the last equation,

aa -bo,= t>r-<*a-

But here ba is identical with the instant of departure /%', and,
consequently,

i.e. the clock placed at a is synchronous with that placed
at b. Q.E.D.

A similar treatment of assumption 2. may be left to the reader,
who will find sufficient hints in Fig. 10. This assumption will be
easily seen to imply that if a pair of flashes be sent out simultaneously

PROPERTIES OF THE SYSTEM <S' 97

from a, one via b^ c and the other via t, b, they will both return simul-
taneously at a. More generally, the time elapsing between the instant
of departure and that of return of the light-signal sent round abca will
be equal to the time elapsing between departure and return of the
signal sent round acba, and similarly for every other closed path in
S, both times being measured by the clock placed at a. This form
of the property attributed to the system S is worthy of being especially
insisted upon, as it implies only operations to be performed at one
and the same spot. To state this property of the system 6", the
observer has not to move from his place.

FIG. 10.

Such then are the physical properties of this system of reference.

It is strange that Einstein, after having made explicitly the above
assumptions i. and 2., considers it necessary to add (p. 894) that
' according to experience ' the quantity

ab

2

ar-ad

or, in the notation of formula (i), the quantity

ab

is to be taken as 'an universal constant (the velocity of light in
empty space).' At any rate, if the last assumption is made, for any
pair of points a, b in 51, once and for ever, then the above state-
ments i. and 2. are certainly superfluous. But considerations of
this order need not detain us here any more.
S.R. G

98 THE THEORY OF RELATIVITY

The properties ascribed to the system S* may be briefly sum-
marised by saying that

Isotropy and homogeneity with respect to light propagation are
postulated throughout S, once and for ever.

In this way the various times, /a, 4, etc., originally foreign to one
another, are all connected so as to constitute one time only, valid for
the whole system, which we may denote simply by /, calling it shortly
the S-time.

There is, thus far, nothing essentially new in Einstein's procedure.
It was more or less unconsciously applied since people began to
measure the velocity of light, and even sound, nay, since they began
to exchange with one another letters or messages of any kind. The
novelty does not come in until the next stage, when the time-labelling
is extended to different systems moving (uniformly) with respect to
one another.

Let ,5" be as above, and let us consider other systems of reference,
S', S", and so on, each having with respect to S a motion of
uniform (rectilinear) translation. Having settled the matter for the
system S, i.e. having established the S-time, /, let us similarly
establish an 6"-time, /', an S"-time, /", etc., and let us see how the
times /', /", etc., are connected with the time / valid for S. It can
reasonably be expected that these processes of (time-) labelling of
events happening at different places, being undertaken from different
standpoints, S, S', S", etc., will generally not coincide with one
another, e.g. that events obtaining identical /-labels may receive
different /'-labels, and so on. Such, in fact, will be the case ; the
labels of different sorts, dashed and non-dashed, though none is
privileged in any way, will have to be carefully distinguished from
one another. In a word, it will appear that, with the above
definition of simultaneity, no universal, no unique time-labelling is
possible.

It will be enough to consider explicitly, besides S, one other
system only, say, S'. Supposing that a consistent time-labelling of
events occurring at different places of S' or an S'-time, is possible,
like the above S-time, the question is, how is this time /' to be
connected with the time /? We shall see that the connexion sought
for will involve also the coordinates defining the position of points

* Which Einstein himself, in order to have a convenient name, provisionally,
calls 'the stationary system.'

EINSTEIN'S PRINCIPLES 99

within S, and within S', In a word, the time-connexion of
both systems will turn out to be entangled with their space
relations.

Here we shall have, to appeal to what Einstein calls the principles
of 'relativity' and of 'constancy of light-velocity,' and wrhich he
enunciates in the following way :

I. The Principle of Relativity. The laws of physical phenomena*
are the same, whether these phenomena are referred to the system S
or to any other system (of coordinates) S' moving uniformly with
respect to it.

II. The Principle of Constant Light- Velocity. Every light-disturbance
is propagated, in vacito, relatively to the system S with a determinate
velocity c, no matter whether it is emitted from a source (body)
stationary in or moving ivith respect to S. The 'velocity' c is
the light-path divided by the corresponding time-interval, c = abjt;
t being the .S-time as denned above.

Here, to begin with and to fix the ideas, the system S is taken.
But applying the principle I., we can say at once that the same
constancy of light propagation is valid also with respect to the
system S'. The constancy of the velocity of light, i.e. its independ-
ence of the motion of the source, as emphasised in Chap. II., has
already been appealed to by Fresnel. But there is this essential
difference that Fresnel claimed this property of light propagation
only for a certain, unique system of reference, namely the aether
or a system fixed in the aether, while Einstein, by accepting I. and II.,
postulates it for any one out of an infinity (°o3) of systems moving
uniformly with respect to one another. With regard to this property
the systems S', S", etc., are perfectly equivalent to the system S or
become so in force of Principle I., — and this is the reason why the
mere notion of an ' aether ' breaks down. None of the systems in
question is privileged. To make it as plain as possible, let P be a
point fixed in the system S, and let a point-source, moving relatively to
S in a quite arbitrary manner, emit an instantaneous flash just when
it is passing through P. Then the observers rigidly attached to
the system S will find that the disturbance is propagated from P
in all directions with the same velocity c, i.e. that the ensuing thin

* Literally : ' The laws according to which the states of physical systems are
changing,' etc. (Einstein, p. 895).

TOO THE THEORY OF RELATIVITY

pulse or wave of discontinuity is a spherical surface, of centre P
and of radius

if / is reckoned from the instant of emission. Again, if P' is a point
fixed in S\ and if the arbitrarily moving source emits a flash just
when it is passing through P\ then the wave, as it appears to
observers rigidly attached to S\ will be a sphere whose centre is
permanently situated at P' and whose radius at any instant of the
»S'-time is

r'~«af,

if /' is reckoned from the instant of emission. Such is, in virtue of I.,
the meaning of the principle of ' constancy ' of light-propagation in
empty space. Of especial interest is the particular case, in which
our source is fixed at a point P' of the system S', and therefore
moving uniformly with respect to S. In this case the centre of
the spherical wave will, to the ,S'-observers, be permanently situated
at the material particle playing, the part of source, whereas for the
^-observers the centre of the spherical wave, fixed in S, will detach
itself from the material source, the source moving away from it
with uniform velocity together with the whole system S'. This
case will be made use of presently.

Let us now return to the first of the above principles, and let
us remember how the time /, valid for the whole system »S, has
been defined. Since S has been endowed with physical properties
required for a consistent method of time-labelling of events occurring
at its various points, the same properties will, in virtue of I., hold
also for S'. Again, local clocks satisfying the requirements of
convenience, e.g. the causality-maxim, being possible in S, such
time-keepers are, by I., possible also for various stations taken in S'.
We can therefore consider first a time /</, measured by a clock
placed at a point a in S ', then distant clocks placed at //, etc.,
leaving the task of testing their synchronism to observers attached
to the system 6", and repeating in fact literally all that has been said
before with regard to the system S. In this way we should obtain
out of the originally local times a unique time t' applicable to the
whole system S '. Let us call the time thus constructed the s'-time.

The question now is, how are the .S'-time and the »S-time con-
nected with one another (and, possibly, with other things, viz.
lengths or distances as measured by the .5"- and ^'-observers) ?

TIMES AND LENGTHS 'COMPARED'

101

The answer to this fundamental question may be obtained, with
the help of the two above principles, in a variety of ways. But for
certain reasons the following way, though not the shortest, seems to
me the most instructive to begin with.* It is, moreover, intimately
connected with what has been said in the last chapter with regard
to the Michelson experiment.

Let us imagine an ^"-observer having at his disposal a point-source
of light at a place P' fixed in the system S'. Let A and B' be a
pair of distant points also fixed in S', and such that the straight line
P'A is in the direction of motion of S' relatively to S, and that
P' B' is perpendicular to that direction (Fig. n). As before, we
shall call P'A' longitudinal, and P' B' transversal. Let /' be the

tB'

xs; o'

/ 0 q \

1 I.. , ' \ '

' length ' of the first of these segments or the ' distance ' from P'
to A, according to the estimation of the .^'-inhabitants, and similarly
s the length of the second segment. Suppose that our observer
sends an instantaneous light-flash from Pf towards A and receives
it back at P' after the lapse & of the /'-time. Then, having assured
himself by any means that an assistant stationed at A sends him
back his signals without any delay, our observer will write

Under similar conditions, if he sends a flash towards! B' and

* Einstein's method of reasoning, as given in his original paper (§ 3, see also
Notes at the end of this Chap. ) may be mathematically interesting, but does not
seem to be the fittest when a clear discussion of the physical aspect of the question
is aimed at.

f To avoid unnecessary difficulties as to hitting the receiving station, now /?'
and now A't it will be best to imagine that our observer sends each time a full
spherical wave of discontinuity or a very thin spherical pulse. This will be
found especially convenient when we come next to consider the same processes
from the ^-standpoint.

102 THE 'TftfcORY OF RELATIVITY

receives it back after the interval r' of the /'-time, he will put down
the equation

25

There is, in fact, by the above principles, no difference between
longitudinal and transversal light signalling between stations fixed
in S', as observed by the inhabitants of this same system.

Let us now see how each of the above two processes will be
described by an observer attached to the system S. Call the lengths
or distances P'A', P B' , as estimated by the ,S-observer, / and s
respectively. Each of these is obtained by ascertaining, with the
help of an appropriate number of synchronous /-clocks, which are
the points of the ,5-system, through which P' and A ', or P' and B'
pass simultaneously, and by measuring the mutual distances of
these points by means of an ^"-standard rod. Similarly, /' and s
are to be considered as the distances P A' and P B' measured by
standard rods which the ^"-observers are carrying with themselves.
Notice that, by Principle I., /' and s', thus measured, will be the
same whether the system S', together with its observers, clocks and
measuring rods, is at rest with respect to S or whether it moves
uniformly with respect to that system, as it actually does. But /, j
are not necessarily equal to /', s'. For although they are ' distances
of the same pairs of material points,' the source and the receiving
stations, they are not obtained by the same processes. Having
thus explained the meaning of /, s, let us consider, from the
^-standpoint, first the longitudinal and then the transversal signal-
ling. The flash sent out by the luminous source will, according
to Principle II., appear to the ^-observers in both cases as a
spherical wave expanding with the velocity c and having its centre
at that point P§, fixed in *S, through which the source has passed
when emitting the flash. Now, if v be the velocity of S' relative
to S, the receiving station A' moves away from PQ with the uniform
velocity v. If, therefore, 01 be the £-time required for the wave
to expand from />0 to A',

and

In the same way, if 0.2 be the ,5-time employed by the light

TIMES AND LENGTHS COMPARED 103

to return from the receiving station* to the sending station />',

Thus, the .S-time 6 elapsing between the first appearance and the
reappearance of a light flash at A', being the sum of 0j and 02,
will be given by

o.*.,^

c- — v

A similar reasoning applied to the case of transversal signalling, in
which case the sphericity of the wave will be found particularly
convenient, will give us for the 5-time elapsing between the appear-
ance of the first and second flash at A' the value

+*<*¥*'(*. j

KN '\f--2iyv--' r=24

&i£j%,.+^ *

Compare the last two formulae with the above ones for & and T',
and denote the ratio s/s' by a. Then the result will be

*& ty-, ^j-* (3) J

where a is a number which for v = o becomes equal i, but is other- ^ J
wise an unknown function of the data of the problem.

Now, each of the two processes, i.e. the longitudinal and the
transversal signalling, may (by disregarding the receiving stations) ^K^
be considered as phenomena consisting in a double appearance of
a flash at one and the same station, at the same individually dis-
cernible point A\ fixed in S'. Thus far we have, purposely, kept
these two processes separate. But now we can advantageously
combine them with one another. If the receiving stations were
chosen so that / = /', then we should have, by the first pair of
formulae,

ff = r', say =T',

and if the two processes were started simultaneously, from the
6"-point of view, they would also have ended simultaneously for
the ^"-inhabitants. In other words, we would have, in S', a pair of

* This station A' (and similarly, in the case of transversal signalling, the station
B') may be imagined to become an instantaneous point-source emitting a spherical
wave at the moment when it is reached by the original wave.

104 THE THEORY OF RELATIVITY

simultaneous events followed by another pair of simultaneous events,
all of these occurring at the same place A'. Let us now require
(what, as far as I know, is tacitly assumed by most authors) that

III. Events locally* simultaneous for an S' -observer should also
be simultaneous for the S-observers.

This amounts to supposing that there is a oiie-to-o?ie correspondence
between the /-labels and the /'-labels to be applied to events
occurring at any given place, i.e. for fixed values of the coordinates
x',y', z in S '. (The analogous one-to-one correspondence between
#', /, z and x, y, z for /' = const, is tacitly assumed as a matter of
course.) On the other hand, two events occurring at distinct places,
being simultaneous in 6", are generally ^^/-simultaneous from the
^-standpoint.

Now, in virtue of the requirement III., call it postulate or
desideratum, or whatever you prefer, the above two simultaneous
processes or phenomena occurring at A will also begin and end
simultaneously for the ^-observers, so that

0 = r, say =T,
and

Consequently, by the equations (3),

These are the required connexions between durations and lengths,
measured in 6" and in S'. They are based on the above assumptions
I., II., III., the last of which is certainly the most obvious. The
common coefficient a is, thus far, indeterminate. If we are to endow
(empty) space with homogeneity, as well as with isotropy,f and if
it be granted that the relations between the S- and ^'-measure-
ments do not vary in time, the unknown coefficient a can depend
only upon v — c^. The only thing we thus far know about this

* i.e. occurring at one and the same place.

fBoth properties having been already attributed to it physically, i.e. as regards
propagation of light, by II.

LONGITUDINAL CONTRACTION 105

function is that it reduces to unity for /? = o, when S' is 'at rest
relatively to S, when, in fact, both systems cease to be discernible
from one another. Thus

a = a(/3), a(o)=l.

Notice that for v = o we have also 7=1, so that in this case T, /, s
become, by (4), identical with T', /', /, as was to be expected.

To put the relations (4) in words as simply as possible, and to
fix the ideas, let us assume for the moment a = i. Then

T I i

Y, = y, ? = -: ,

for a - / ,

Thus^,' a transversal bar" sharing the motion of S' will have the same
length from the standpoint of either of the two systems S, S', while
a bar of longitudinal orientation and of length /' in S' will, according
to the estimation of the ^-observers (with equal /-values for both
terminals of the bar), be shortened to /=/'\/i -/?"• A solid fixed
in 6", which for the inhabitants of that system is a sphere of
radius R, will, according to the estimation o£ the ^-observers, become
a longitudinally flattened ellipsoid of semi-axes

-R, R, 7?,

precisely as in the contraction hypothesis of Fitzgerald and Lorentz.
It is a slightly different thing to say, instead of this, that a body
which for the S-observers is spherical while at rest in ,S becomes
flattened down to the above ellipsoid when set in motion with the
translation-velocity v relative to S. The clause hinted at is in
'connexion with the manner in which the body is set from rest to
motion and cannot satisfactorily be dealt with at this stage of our
considerations. Again, as regards the ratio of times, remember
that T' is the ^"-duration of a phenomenon or process going on at
a place P' fixed in S\ i.e. for constant x, y, z. This duration or
time-interval is then lengthened in the estimation of the ^-observers
to T=yT'= T'fji -fi*. We are assuming here, of course, that
ft< i, so that 7 is real and greater than unity. Instead of a pair of
flashes, as considered above, we may think of two consecutive indica-
tions of an S' clock preserved at P' , and we may say that a clock
moving relatively to S with the uniform velocity v goes slower, in the

106 THE THEORY OF RELATIVITY

ratio \/i -/32 : i, than 'the same ' clock when at rest in S. This at
least is the way that the leading relativists put the above result.

* The same ' is taken to mean that the mechanism of the clock has
undergone no changes due to its passing from rest to motion,
except those which are implied by the fundamental relativistic prin-
ciples themselves. This statement does by no means look satis-
factory, but it can be made more rigorous and clear by returning
to it after certain portions of relativistic physics have been
worked out. The practically important question is, which are the
physical systems we are going to consider as such clocks whose

* internal mechanism ' is not subject to changes due to their merely
passing from rest to uniform motion relatively, say, to the earth or
the fixed stars. Now, as far as I know, the prevailing tendency is
to consider as such physical systems the various atoms (or at least,
if they are to serve us for thousands of years, those which are
not sensibly radioactive) with their 'natural' periods of vibration,
manifested in their characteristic spectrum lines.* The influence
felt by such minute mechanisms in the presence of a strong magnetic
field (Zeeman's effect) will not, of course, be forgotten. Who knows
but that some remote future generations, to get rid of such physical
influences, may choose to consider as 'invariable' the mechanism
not of light emission but of radioactive disintegration of atoms.
If such is to be the case, the formula T=yT' will be interpreted
by saying that the 'half-life' of radium, which is about 1760 years,
is in the estimation of a terrestrial observer lengthened by a month
or so, when flashing before him with something like one hundredth
of the velocity of light.

We have already remarked in passing that two events occurring
simultaneously in S' at places distant from one another will generally
be non-simultaneous to the »S-observers. This may be seen im-
mediately by the principle of constant light-velocity, valid by I. for
both S and S'. For let a spherical wave or a very thin pulse be
started from our point-source placed at P' . Then, if l' = s't the
arrivals of flashes at A' and B' will be a pair of events simultaneous

*Thus we read, in M. Laue's Relativitiitsprinzip, second edition, 1913, p. 42 :
'In einem bewegten Wasserstoffatom (Kanalstrahlen) werden, zum Beispiel, die
Licht emittierenden Eigenschwingungen geringere Frequenz haben, als in einem
ruhenden.'

As regards the experimental side of the subject, see J. Laub's report in
Jahrb. d. AW. u. Elektronik^ Vol. VII. p. 439.

RELATIVITY OF SYNCHRONISM 107

to the ^'-observers. On the other hand, the ^S-time required for
the wave to reach A' will be

f , _

and that to reach B'

' c
Now, by (40), and also by the more general formulae (4),

whence

T _T = @7S / / ~J.^*rtVfcfc/*r

™ c ' (ft* /*' r*-f*&

Thus, the pair of events in question will not be simultaneous for
the ^"-observers. Instead of the two particular points A', B', the
whole wave may be considered. Then it will be seen at once that
the sphere r = const, with centre P' will, to the ^"-observers, be the
locus of points reached simultaneously by the wave, but not so to
the .S-observers. For to these the loci of simultaneously illuminated
points will be spheres centered at a point, PQ, fixed in S, from
which P' is continuously moving away.

Thus, the notion of distant simultaneity, to call it again by this
short name, has no ' absolute ' or universal meaning, but involves a
specification of one out of oo3 systems of reference. For such is
the manifold of the vector-values of their relative velocity v, its
absolute value v amounting to one scalar, and its direction to two
more. T ^ A k' S ^O 3. M ATI Q M

Let us now once more return to our formulae (4), with the view
of deducing from them the relations connecting the *S-time and
coordinates /, x, y, z with the S'-time and coordinates /', x\ y, z'.t
Take the #'-axis coincident in direction and sense with the o:-axis,
both concurrent with the vector v fixing the velocity of S' relative
to »S (Fig. n),* and the axes of y, z, both transversal and per-
pendicular to one another, parallel to and concurrent with the axes
of y, z respectively. Count both the S'- and the »S-time from the

* In that figure the systems S', S are represented as sliding along one another
only to avoid confusion in the drawing, but in reality they are to be imagined as
interpenetrating one another throughout the whole (three-dimensional) space.

loS THE THEORY OF RELATIVITY

instant at which the origins of the coordinates, O and O, coincide
with one another, i.e. assume

as corresponding to

which is a pure convention. The axes of y and z will then coincide
at that instant with the axes of y and z. Lev us fix our attention
on any point P'(x, y', z'} taken in S'. Then by the second of
formulae (4), in which we have to write l=x — vt, l' = x,

*'-£(*-«"), (5)

and, by the third of those formulae,

y' = -y- z' = -z. (6)

J a/ ' a

To obtain t as a function of x\ y , z'9 /', notice first of all that
events occurring at various points of a transversal plane (x — const),
being simultaneous in S', are also simultaneous with one another
according to the 5-point of view. For if M', N' be a pair of such
points, and if M'N' — s\ then a wave started at their mid-point C'
at the instant t' - \s\c will reach both M' and N' simultaneously,
at the instant /'. Again, from the -S-standpoint, in our previous
notation, , ./ ,

^ Ty S

J'c. , ff y$

2C

so that M' and N' will receive the signals at the same instant /.
Thus, / is independent of y', z', and consequently

t=t(x', t').

Next, take a longitudinal pair of points, say P' on the .r'-axis and
the origin O'. Call x the abscissa of P'. Imagine a wave started
at the mid-point of O' and P' at the instant /' - \x'lc\ then the wave
will reach O and P' at the same instant /', and, by Principle II.
and by the second of formulae (4),

,,,, , ,, a x' / i i\ v t
y 2 \t — v c+v) •*-'<*"

THE LORENTZ TRANSFORMATION 109

But, by the first of formulae (4) and by the above convention as
to the origin of time-reckoning at O',

/(o, t') = ayt'.
Hence

(7)

which is the requir^I connexion. Substituting here x' from (5)
and remembering that /324- i/y2= i, we shall obtain /' in terms
of/, A-. * *A*4+fy*ty m*4tt£*fa~4+ig-/

Thus, the complete set of formulae connecting the S'- with the
.S-time and coordinates will be

(8)

Conversely, resolving these equations with respect to /, xt y, z, or
simply copying (7) and using it to eliminate t from the first equation,

x — ay (x + #/'); y = ay' • z — az'

(9)

-«y(t' + ~*').

Notice that, disregarding a, the set (9) follows from (8), and vice
versa, by simply interchanging x, y, z, t with x', y', z', t' and
by writing -v instead of v. Now v being the velocity of S'
relative to S, -v will be the velocity of S relative to S'.* As
to <:, it is common to both systems, and y (v) = y( - v) = (i - v2/c2)~^.
Thus, there is reciprocity between the two systems of reference,
except for the common arbitrary coefficient which is a-1 in the

* In fact, what we call the velocity of S relative to S' is the vector whose
components are the derivatives of x', y', z' with respect to t',for constant x,y, z,
that is to say, by (8),

dx' _ dy' _ dz' _

and this is the vector - v. In exactly the same way, the velocity of S' relative to
.V is the vector whose components are the derivatives of x, y, z with respect to /,
jor constant x' ', y', z', i.e., again by (8),

dx _ dy _ dz _

no THE THEORY OF RELATIVITY

first, and a in the second set of formulae. As a matter of fact,
there is a physical reciprocity anyhow, i.e. for any a = a(£>), subjected
to the condition a(o)=i. For the conditions imposed' upon the
time-labellings in S and in S', in order to make them self-consistent,
will continue to be satisfied when all values of time and coordinates,
in S or in 6", have been multiplied by a common factor ; or1 in one,
and a in the other case may be thrown back upon the choice of the
units of measurement. Thus, the choice of a being a matter of
indifference, we may take a = i. But, if not content with the physical,
we require also a formal reciprocity, then we have to write

a"1 = a, i.e. a? — i.

But a(o) = i. Thus, if a(v) is to be continuous, a= + i.*

In this way we obtain the formulae of what is universally called
the Lorentz transformation,

x' = y(x-vt); y=y; z' = z

already met with in Chap. III. But here, as can be judged from
the whole line of reasoning, the meaning and the role of this
transformation are essentially different from what they were in
Lorentz's theory, based as it was on the assumption of a privileged
system of reference, the aether.

Let us write also the inverse transformation

t=

y=y • z=

(10')

The above postulate I., or the Principle of Relativity, may now
be expressed in the concise and more definite form :

I". The laws of physical phenomena, or rather their mathe-
matical expressions, are invariant with respect to the Lorentz
transformation, f

*With regard to Einstein's own treatment of this subject, and also that
adopted in Laue's book, see Note 1 at the end of the present chapter.

t Some authors employ in this connexion the mathematically sanctioned term
covariant, instead of invariant. But it will be convenient to reserve ' covariant '
for another use, namely to denote that two groups of magnitudes are equally
transformed.

THE PRINCIPLE OF RELATIVITY rn

That is to say, if a law Z, valid in S, involves — besides other
magnitudes — x, jr, z, t in a certain way, and if these are transformed
according to (10), then the resulting law Z', valid in 6", will involve
A-', r', z, t' in exactly the same way. Any system S\ with its
corresponding tetrad of independent variables, is as ' legitimate '
as S. The choice of one out of co 3 systems of reference moving
uniformly with respect to one another is a matter of indifference.
As regards the behaviour of the * other magnitudes ' involved in the
laws, any attempt to elucidate it by general remarks in this place
would be useless. We shall come to understand this point by and
by when considering various applications of the above principle.
And, with regard to the specification ' physical,' it has, of course, to
be taken in the broadest sense of the word. The phenomena in
question may as well be chemical or physiological (though, for the
present, physiology is far from being prepared to receive a theory
of such a high degree of accuracy). Instead of ' physical phenomena '
the reader can, at any rate, put theoretically : any phenomena which
are at all localizable in space and in time. But subtleties of this
kind need not detain us here any further.

The principle of relativity excludes all such laws as are not
invariant with respect to the Lorentz transformation. Thus, for
instance, Newton's inverse square law of universal gravitation, or
even his general laws of motion, cannot stand in their original
form, but require some slight modifications, if they are to be
brought into line with the principle in question. But there is
certainly no need to multiply such negative examples ; the reader
can pick out at random as many cases as he wants, and he is
sure never to hit a case which does not contradict the principle of
relativity. Maxwell's equations for the 'free aether,' also with the
supplementary term pp, and for ' stationary ' ponderable media, are,
as has been already remarked, in an exceptional position. But these
electromagnetic equations will occupy our special attention in later
chapters.

Thus far we have had only one example of a ' law ' which is
proclaimed to be rigorously valid, with reference to S, namely the
law of light propagation, as enunciated in the principle of constant
light-velocity.* Thus, the true office of II. is to fix a particular
case of a physical law which is postulated rigorously to satisfy I.

* Notice that, in considering this law, we need not yet trouble about the
electromagnetic, or any other, theory of light.

U2 THE THEORY OF RELATIVITY

This law then has certainly to be invariant with respect to the
Lorentz transformation. And since this transformation has been
obtained by means of the law itself, applied both to £ and S', it
can be foreseen without calculation that this law will prove to be
invariant. In fact, this prevision may be verified at once. For the
law in question states that if light be emitted at the instant / = o
by a point-source, placed at or just passing through a given point,
which may be taken as the origin of the coordinates, O, then at
any instant />o it reaches a spherical surface of radius r — ct and
centre O, i.e. such that, x, y, z being the coordinates of that surface,

Now, squaring the equations (10) and adding up, we have, identically,
x'* +/2 + zv _ &'* = *a +y* + z*_ &*, (12)

and consequently also

Thus, the law of light propagation, (n), is invariant with respect
to the Lorentz transformation. Remember that O' coincides with
O for /=o, when also /' = o, and that, therefore, (u') expresses
for S' precisely the same thing as (n) for S. Notice, moreover,
that the law under consideration would be invariant with any value
of a (not zero). For, then, we should have, by (8),

and what we require is not so much the invariance of the quadratic
function x2 +y2 + z2 - <r2/2 as that of the equation (n). But having
once decided, be it only for purely formal reasons, to take a=i,
the property (12), which will in the sequel be often referred to, is
worth keeping in memory.

It may be expressed shortly by saying that

a-2 +y2 + z2- c2t2, or r2 - W

is a relativistic invariant. Any function of this expression alone is,
of course, again an invariant. But all of these count as one
invariant. It is worth noticing that, on this understanding, there
are among all functions of x, y, z, t no other invariants than

x- +2 + z2 - c-f1.

INVARIANCE OF DALEMBERTIAN 113

In what precedes we have used the integral form, (n), of
(a particular case of) the law of propagation. We might as well have

used its differential form,

O£ = o, (13)

where <£ may be thought of as any one of the rectangular com-
ponents of a 'light-vector,' and where

I 3* B2 d2 92 , B2

n = V~-?> W = & + ty* + W~c* W

is Cauchy's symbol, called also the Dalembertian. The physical
meaning of this famous differential equation is (among other things)
that any element of a wave of discontinuity is propagated normally
to itself with the velocity c (cf. Note 2). This then is the general
law of which the previous is but a particular case, corresponding
to a particular form of the wave. Now, by (10),

_

7dx

so that

n=a, (15)

which proves the invariance of the differential law of the propagation
of light in empty space. But since (13) involves further particulars
not yet entered upon (embodied summarily in <£) concerning light,
the reader is recommended to keep rather to the above integral
form (u), until we come to consider the relativistic properties of
electromagnetic laws. Meanwhile he is asked to retain in memory
solely that the Dalembertian is an invariant as good as r^ - c*-tf
although the latter is a magnitude and the former an operator.

Conversely, the Lorentz transformation may be obtained by
postulating the invariance of the Dalembertian and by making some
auxiliary assumptions (Note 3). But the above method of obtaining
the transformation formulae seemed to me to be more suitable for
bringing into prominence their physical meaning.

Basing ourselves upon the Principles I., II., and upon the

obvious requirement III., we have obtained the formulae (40) for

the ratios of time-intervals and lengths as measured in S and S'.

From these formulae the Lorentz transformation (10), and its inverse

S.R. H

H4 THE THEORY OF RELATIVITY

(10'), followed almost immediately. Now, it may be well to notice
here how (40) are to be obtained conversely from (10), (10'). The
third of (4^) is identical with y =y', z = z. To obtain the first of
formulae (4^), remember that jt, was valid for a point (any point)
fixed in S'. Take therefore, in the last of (10'), x' = const., and
denote by A any increment. Then the result will be

Similarly, remembering that the terminals of the segment / are to
be taken simultaneous in S, take, in the first of (10), / = const. ;
then the result will be

A* = -A*'.

y

Now, these are precisely the relations stated by (40}. Notice that
the constancy or variability of the transversal coordinates jr, z is
a matter of indifference. As to the fact, mentioned on several
occasions, that simultaneous events occurring at distant places in
S' are generally not simultaneous in S, and vice versa, it is most
immediately expressed in (10), (10') by the circumstance that
t contains x besides /', and similarly, that /' contains x besides /.
So long as v<c, or /3 < i, the coefficient y is real and greater
than unity, so that the duration of any process, local in S', is
lengthened, to the ^-observers, y times or in the ratio i :(i -/22)Y,
and any longitudinal segment A#' is contracted to A#'(i -y82)^. In
the critical case of v = ct or (3= i, we have y = oo. Then any finite
duration A/' becomes infinite in S, and any finite distance A#', as
judged by the .S-observers, dwindles down to nothing : the whole
of S'j with all the bodies sharing its motion, becomes a transversal
flatland. Finally, for /3>i, i.e. when the velocity of S' relative
to S exceeds the velocity of light or when it becomes what may
conveniently be called a hyper-velocity, * y is purely imaginary and so
also are x, t for any real values of x', t'. But, as far as I can see,
this does not necessarily mean that motion with hypervelocity, of
one body relative to another, is 'impossible.' It would, thus far,
be enough to say simply that there is in this case no correlation
in real terms between S' and S to be obtained by light-signalling.
Notice that, from the 5-standpoint, any station P' can then succeed
in sending light-signals only to points contained in a certain back-

*The Germans call it ' Ueberlichtgeschwindigkeit.'

OLD AND NEW RELATIVITY 115

cone, so that, according to that standpoint, no such station can ever
receive back any of its signals, and that therefore the whole of our
previous reasoning ceases to be applicable to the case in question.
In what sense hypervelocities are, or by what reasons they may
be required to be, 'impossible,' will be seen from the physical
applications of the principle of relativity.

For the present, and for what follows, we shall simply assume

v<c,

considering only now and then the limiting value v = c.

To touch the other extreme, let us suppose that v is a very small
fraction of c. Then, neglecting /?"2-terms, and limiting ourselves
to such values of x as are not enormously great compared with
ct, we obtain from (10) the Newtonian transformation (Chap. I.)

Or, if we like, we can say also that, if oo is taken instead of r, the
Lorentz transformation reduces to the Newtonian transformation.
Just as the equations of classical or Newtonian mechanics were in-
variant with respect to the Newtonian transformation, so are the
fundamental laws of optics and (as we shall see later) of electro-
magnetism invariant with respect to the Lorentz transformation.
Let us call the principle associated with the former the classical
principle of relativity, and that corresponding to the latter of these
transformations the new principle of relativity. Then it is obvious
that we cannot have both, retaining the classical principle for our
mechanics and using the new one for our electromagnetism. For
if S be a particular system or space-framework of reference in which
the laws of both classical mechanics and electromagnetism are valid,
then, among all the systems moving with respect to it with uniform
velocity, no other would have this property.* In other words, the
system S would be privileged, being the system for both classes
of laws, whereas, according to the general principle of relativity,
i.e. according to I. taken by itself (without yet touching II.), none
of the manifold of cc3 systems moving uniformly with respect to
one another is to be privileged, equal rights being claimed for all
of them with regard to any physical phenomena. Thus, if we are

* Supposing, of course, that the inhabitants of each system avail themselves
only of one set of coordinates and time.

n6 THE THEORY OF RELATIVITY

to construct a truly relativistic theory at all, we can have but one
Principle of Relativity, that is to say, one at a time. (It may well
happen that the next, or even the present, generation will have to
give up the 'new' principle for a yet broader one.) Now, Hertz's
and Heaviside's attempts to extend the classical principle of relativity
to the domain of electromagnetism proved a complete failure. And
since, for the time being, tertium non datur, the 'new' principle,
involving the Lorentz transformation, has become the principle of
relativity of modern physics. In this connexion it must not be
forgotten that electromagnetic and especially optical phenomena
have been known all these years with a much higher degree of
accuracy than the various instances of motion of material bodies.
No wonder, therefore, that the physicist has so easily decided to
mould his mechanics and thermodynamics according to a principle
which sprang out from optical and, generally, electromagnetic ground.
This is not to say, of course, that mechanical and all other pheno-
mena must be ' ultimately ' electromagnetic, i.e. that everything must
be explained by, or reduced to, electromagnetism. The theory of
relativity is not concerned at all with such reduction of one class
of phenomena to another. It does not force upon us an electro-
magnetic view of the world any more than a mechanical view. Quite
the contrary; it opens before us a wide field of possibilities of
asserting that even the mass of a free electron, say a /2-particle, must
not be entirely electromagnetic.

Like the Newtonian transformations, the Lorentz transformations,
generally with the inclusion of pure space-rotations,* constitute a
group, that is to say, two of such transformations applied successively
one after the other are equivalent to a single transformation, which
is again a Lorentz transformation. In the case of the Newtonian
transformation, if vx be the velocity of S' relative to S, and V2 the
velocity of S" relative to S', the vectorial parameter v of the
resultant transformation is simply the sum of the parameters of
the component transformations, i.e. V=v1-fv2. The parameter
of the resultant Lorentz group is a more complicated function of
the parameters of the component transformations, "thus leading to a
more complicated rule of physical addition of velocities, which will
be given' in the next two chapters. Only wrien the absolute values
of vls V2 are small compared with the critical velocity, does the
familiar rule of composition of velocities reappear. Classical kine-
* This reservation will become clear in Chapter VI.

EINSTEIN AND LORENTZ 117

matic is but a limiting case of modern relativistic kinematic. So
are also most of the remaining branches of mechanics and generally
of physics. For slow motion the reader will recognise throughout
his good old friends in this new and strange land of relativistic
connexions.

To close this somewhat lengthy chapter on the foundations of
the theory of relativity, one short remark more. Einstein's results
concerning electromagnetic and optical phenomena will be seen
to agree in the main with those which have been obtained by
Lorentz in his generalized theory, the chief difference being (to
quote Lorentz's own words, Columbia University Lectures, p. 230)
that Einstein simply postulates what Lorentz has deduced 'with
some difficulty, and not altogether satisfactorily, from the fundamental
equations of the electromagnetic field. By doing so, he may cer-
tainly take credit for making us see in the negative result of
experiments like those of Michelson, Rayleigh and "Brace, not a
fortuitous compensation of opposing effects, but the manifestation
of a general and fundamental principle. ... It would be unjust
not to add that, besides the fascinating boldness of its starting point,
Einstein has another marked advantage over mine. Whereas I
have not been able to obtain for the equations referred to moving
axes exactly the same form as for those which apply to a stationary
system, Einstein has accomplished this by means of a system of
new variables slightly different from those which I have introduced.'
(As to these slight differences, cf. Note 86 to Lorentz's Lectures?)

We see from the above quotation that Lorentz himself aimed at
an exact sameness of form of the laws of all, or at least of electro-
magnetic, phenomena for a pair of systems moving uniformly with
respect to one another. Why then not postulate this sameness
at once? But Lorentz had not the heart to abandon the aether
which he confessedly ' cannot but regard as endowed with a certain
degree of substantiality.'

ii 8 THE THEORY OF RELATIVITY

NOTES TO CHAPTER IV.

Note 1 (to page no). Einstein, Ann. d. Physik, Vol. XVII., 1905, §3,
obtains the formulae of transformation (10) in the following way :

Let, to use our notation, x, y, z^ t be the coordinates and the time in S,
and x', y', z', /' those in S'. Write

then to a point fixed in S' corresponds a system of values of £, y, z
independent of /. To obtain /' as a function of £, y, z, Einstein considers
a signal sent at the instant /0' from the origin O' along the axis of x'
towards the point £, where it arrives at the instant /1? and, being reflected
there, returns to O' at the instant /</. Then, according to the definition
of synchronism, (\a\ p. 95, which is to hold equally for S' as for 5,

i.e. filling in the arguments and applying the principle of constant light
propagation,

f(o, o, o, /)+/'<>, o, o,
whence, for an infinitesimal £,

-

Applying the same reasoning to signals sent along the axes of y or z,
Einstein obtains

-d? 'dt'

and, assuming /' to be a linear function of its arguments,

where </>(^) is thus far an unknown function of v, and where /'=o has
been put at O' for /=o.

Next, to obtain from the last equation _r', y' , z' in terms of x, y, z, /,
Einstein writes the principle of constant light-propagation in S'. A
signal started at O' at the instant t' = o reaches at the instant /' a point
of the positive .r'-axis, for which

THE LORENTZ TRANSFORMATION 119

But the same process, if considered from the S-standpoint, gives
£ = t(c-v). Thus

*-tt*)?r

Similarly

S = ct' = 4>(v}.

where t=y(c--v*)~*, £=o. Thus

and z' = <l>(v)yz.

Consequently, writing again ^=.r — •z//, and throwing the common
factor y upon <£(z>),

These are identical with the formulae (8) of the present chapter, for
<j>(v)=i/0(, The way that Einstein obtains the particular value <j>(v)=\
(loc. tit. pp. 901-902) need not detain us here. We know that the value
of such a common coefficient is essentially, from the physical standpoint,
a matter of indifference.

As to Laue (Das Relatimtdtsprinzip^ 2nd edition, p. 38, etc.), his
method of obtaining the Lorentz transformation consists in postulating
the invariance of the ' wave-equation ' .

and in assuming linearity and symmetry round the axis of motion,
i.e. in writing

where K, A, /A, v are functions of v alone. These functions are then easily
determined from the postulated invariance which Laue writes

where a is again an unknown function of v alone. The value of A is
easily shown to be equal to unity, by requiring reciprocity, i.e.

120 THE THEORY OF RELATIVITY

and by remembering that ' for the y- and ^-directions it is exactly the same
thing whether S' moves relatively to 5 in the positive or in the negative
sense of the .r-axis,' so that A(7/) = A( — v). Thus y'=y, z' = z, and, by

/ 7/2X-5- v

(b\ K = fji = ( i — 5J = y, v = —1y. Substituting these values in (a\ Laue

obtains the required formulae (10). The discussion of Laue's method of
obtaining for a the particular value I, rather than any other, is again
left to the reader.

Note 2 (to page 113). Let the function <£, satisfying the equation
n$ = 0> be continuous, as well as its first derivatives
etc., that is to say, let

but let the derivatives of the second order, c32<£/9/2, 32c£/d.r2, etc.,
experience a discontinuity at the surface cr. Then, TL = i7t1+jn2+]s.n3
being the normal of any surface-element dor, at the instant /, the identical
conditions and the kinematic conditions of compatibility, expressing that
a- is neither split into two or more surfaces, nor dissolved, at the next
instant t+dt, are (cf. Ann. der Physik, Vol. XXIX., 1909, p. 524)

where b is the velocity of propagation, along n, and A a scalar
characterizing the discontinuity. Now, n being a unit vector,

[V2<£] = A, and

whence |b| = <r. Q.E.D.

In electromagnetism (f> has in turn the meaning of the components
of the electrical and the magnetic vector, and the sense of propagation,
±n, follows from the mutual relations of these two vectors.

Note 3 (to page 113). Postulate the invariance of the Dalembertian, i.e.

Q=D,

and assume

or make any set of plausible assumptions leading to this. Then

and

THE LORENTZ TRANSFORMATION

Instead of x, t introduce new independent variables

121

and similarly, for the system 5',

Then the required invariance will assume the form

w

Now, considering £', T/' as functions of £, ->;, without assuming their
linearity, we have

and

Thus, by (a\

^ V

/ -"V / O^/ O /

On , Oh Of}

- •*sf'i"!2?j? ^r~

g Og Or)

To satisfy the third of these conditions, put

then the fifth will become

so that the only possibility of fulfilling the fourth condition consists in
taking

Thus,

£'=£'<£ ^'=V

122 THE THEORY OF RELATIVITY

Hereby the first and second of the above conditions are identically
satisfied, and the fifth becomes

[An alternative solution would be 3£'/3£=o, and 'dr)'j'drj=o, i.e. £' = £'(r)),
97' =?/(£), with (dg'/drj).(dr)'/dg)=i ; but this may easily be shown to
lead substantially to the same final result as the above one.] Now, for

x=vt=c$t,

we require .r' = o, i.e.

for every / ; hence, differentiating with respect to /", and supposing v
constant,

and, by (b\

where both square roots are to be taken with the same sign, namely the
positive (since £' = £, etc., for /? = o). Here ( ), in the differential coefficients,
means 'for x=vt' ; but since £', 7;' depend only on £, T; respectively, these
formulae are valid for any arguments. Hence, integrating, and remember-
ing that for x = t=o, i.e. for £ = 17 = 0, we require g' = r)' = o,

This intermediate form is worth notice, since it shows at once that

Introducing again the values of £, etc., in terms of xt etc., (c) are readily
seen to be identical with the required formulae

CHAPTER V.

VARIOUS REPRESENTATIONS OF THE LORENTZ
TRANSFORMATION.

PASSING now to consider the various expressions of the Lorentz
transformation, which was seen to be fundamental for the whole
theory of Relativity, let us first of all deprive the X-SLXIS of its
(formal) privilege and write (10), Chap. IV., symmetrically in x, y, z,
or, using vectors, avoid splitting into Cartesians altogether. This is
done in a moment. In fact, remembering that our axis of x was
longitudinal, and those of y, z transversal, and calling r the vector
drawn from O to any point in S, and r' its ^"-correspondent, we
can write the first of (10),

where i is the unit of v, similarly the second and third,

r'-(r
and, finally, the last of (10),

To obtain the full vector r' combine its transversal and longi-
tudinal parts, and to get rid of the new letter i, write (ri)i = (rv)v/z;2.
Thus, the concise vectorial form of the Lorentz transformation,
exhibiting its independence of the choice of coordinate axes, will be

124

THE THEORY OF RELATIVITY

Here v is the velocity of S' relative to S, and y = ( i - /32) \ /3 = v/c,
as before.

To suit the non-vectorial reader we may again split (i) into
Cartesians. But in doing so, let us this time take any set of
mutually perpendicular axes x, y, z, for S, which are also to be the
axes of #', y, zt x"9 y", z", etc., for all other systems S', S", etc.,
moving uniformly with respect to one another. Call vx, vy, vz the
components of v taken along these universal, but quite arbitrary,
axes. Then, projecting the first of (i) upon these axes and re-
writing the second of (i), the required symmetrical form will follow,
viz.

where (rv) may be looked at as an abbreviation for xvx+yvy + zvz.
The inverse transformation is obtained by transferring the dashes
from x, y', z', t' to x, y, z, t, and by changing the sign of v, that is

Of VX, Vy, VZ.

On the other hand, to condense the vectorial form (i) still a
little more, observe that r enters into the first of (i) by the

expression r + .2 v(vr) only. Introduce therefore the linear vector
operator

. (2)

Then the Lorentz transformation will be expressed by
r' = er - vy/

Write again, for a moment, v/z; = i, and let j, k be a pair of unit
vectors normal to one another and to v. Then (2) may be written
i -i(i, or, i being the 'idemfactor,' i.e. i(i+j(j+k(k,

THE LORENTZ TRANSFORMATION 125

This is called a dyadic.* Considered as an operator it is a
symmetrical linear vector operator, so that if A, B be any pair
of vectors

(A.eB) = (B.«A). (3)

But the operator e may be described most immediately by calling
it a longitudinal stretcher, since it stretches or magnifies 7 times any
longitudinal vector, i.e. any vector parallel to v, and leaves unchanged
any transversal vector. According to the usual terminology, 7 would
be the ratio of this stretcher.

Observe that v enters into e through 7 only, i.e. quadratically.
Thus, the inverse transformation will be

The above form of the Lorentz transformation, involving (one
vectorial parameter v or) three scalar parameters vx, vy, vz, is
especially useful when there are more than two systems, S, S', S",
to be considered, and when the velocity of S" relative to S' is not
parallel to that of S' relative to S.

But before proceeding further let us yet dwell a little more upon
the properties of the sub-group contained in (i^), which involves
one scalar parameter only, and which covers the particular case of
parallel velocities. This case is especially interesting and instructive
as illustrating a fundamental theorem of Lie's theory of groups of
transformations f and as preparing the way for a subsequent form
of the Lorentz transformation, adopted for illustrative purposes by
Minkowski.

Measuring x, and x', along the direction of motion of S' relative
to S, write again, as in the last chapter,

y =y, s' = s,j

*Cf. for instance my Vectorial Mechanics, London, Macmillan & Co., 1913,
p. 97. The dots used there as separators are here replaced by ( . Thus er means

t Theorem 3 in Vol. I. of S. Lie's Theorie der Transformationsgruppen,
Leipzig, 1888, p. 33. See also the whole of ' Kapitel 3. .fiVwgliedrige Gruppen
und infinitesimale Transformationen,' Ibidem, p. 45.

% That these transformations form a group, and that therefore Lie's theorem
must be applicable to them, is easily seen. In fact, if vl is the velocity of S'

126 THE THEORY OF RELATIVITY

and differentiate x\ y, z', t' with respect to the parameter v. Then,
denoting dyjdv by 7,

dx 7 , dt' . / v \ 7

and using the inverse transformation x = y(x' +vt'), etc.,

dv dv

To see that this is precisely the form corresponding to Lie's theorem,
which, writing a instead of v, and #/(* = i, 2, 3, 4) for x', y', z, t ',
would be

~ = fa (a) . &«, xz', x3f, O, (6)

we have to remember only that y2 = (i -/J2)"1, P = v/c, so that

and consequently

7/7 -

relative to 6", and z;3 that of S" relative to S' (v.2 being taken from the 5"-point ot
view and z/x from the 6"-standpoint), then we have

and

x" = y»(x'-v>2t'}, r*=

and substituting the first in the second, we obtain at once

y" =y, z"

which is again a Lorentz transformation like each of the above ones, namely
with the parameter (velocity of S" relative to S)

This formula embodies the simplest case of Einstein's ' addition-theorem ' of
velocities, which will occupy our attention in the next chapter.

MINKOWSKI'S EXPRESSION 127

identically. Thus, the differential equations (4), (5), with the
omission of the obvious dy'ldv = dz'ldv = o, become at once .

or, writing

l' — ict\ and similarly l=ict, (8)

where i = <J — i,

~dv

« y , (9)

-=- = -i—x.

dv c

Here, the coefficient on the right side being in both equations the
same known function of v, the idea easily suggests itself to introduce
instead of v the new parameter

w = arc tan (i/3) . ( i o)

With this new variable the above equations become
dx .. dl'

Using the well-known general integral of these simple equations
and remembering that for j3 = o (i.e. for o> = o) x' = x, /' = /, we
obtain the remarkable expression of the Lorentz transformation :

4-/sina>; y =y\ z' = z\

(n)
' = /cos w-a:sin(u, J

which was first given by Minkowski, who made it his starting point.

Thus, the Lorentz transformation may be described as a
rotation, in the four-dimensional space x, y, z, /, through an
imaginary angle w in the plane x, /, or ' round the plane ' y, z.

* H. Minkowski, ' Die Grundgleichungen fiir die elektromagnetischen Vorgange
in bewegten Korpern,' Got finger Nachrichten, 1907 ; reprinted in 'Zwei Abhand-
lungen liber die Grundgleichungen der Elektrodynamik,' Teubner, Leipzig, 1910,
p. 10.

128 THE THEORY OF RELATIVITY

That the transformation in question is a pure rotation, i.e. without
•change of 'length,' (x^ +y2 + z'2 + 1'2)^, is best seen from (90), which
give at once

showing thus the invariance, already noticed, of x* + /2, and con-
sequently also of x2 +y2 + z2 + 1'1. Notice that the above rotation
a) is an imaginary Euclidean rotation in x, y, z, /, or, which is the
same thing, a real non-Euclidean (Lobatchewskyan) rotation in
the space x, y, z, ct through an angle ^ connected with o> by

tan w = i tan ^. (12)

We shall soon have an opportunity to return to this real angle,
which, according to (10), is defined by

tan^ = /J. (13)

Let again Vj be the velocity of S1 relative to S, and V2 that of S"
relative to S', the former from the S- and the latter from the ,S'-point
of view. Then, if v3 and V9 be parallel to and, say, concurrent
with one another, the corresponding rotations are

(DI = arc tan (i/^)

round a certain plane, in the four-dimensional space x, y, z, /, and
o>2 = arc tan (i/?2)

round the same plane. (In three dimensions the rotation is round
an axis, or line, in four ' round a plane,' i.e. leaving fixed a whole
plane instead of a line.) Thus, the resultant rotation, corresponding
to the passage from the S- to the ^"'-variables, will be

Not the velocities themselves are added but the corresponding
angles of rotation.

To verify the last formula, call v = cfi the resultant velocity, corre-
sponding to w. Then

£ _!_/?

i/3 = tan CD = tan (wj + w2) = / —
or

THE LORENTZ SUB-GROUP 129

Now, this is but a particular case (cf. footnote on pp. 125-6) of
Einstein's general formula for the composition of velocities, to be
fully considered later on.

Since the sub-group under consideration contains the identical
transformation, namely for V = Q or w = o, it must be possible,
according to Lie's Theorem 6 (loc. dt. p. 49), to represent it as a
'group of translations] i.e. by

In fact, by (go) we have the simultaneous system

dx dl' . ,

~f- = -- T = dw j ay = dz = o,
/ x

with the initial conditions x' = x, y =y, z' = z, /' = /, for w = o.

Whence

x'* + t'* = x* + !* = <!>£ say,
and

dl' ,

Thus, we have only to write

<t>= <£ = s <£ = arc sn

and the Lorentz transformation will assume the required canonic
form

<k' = <k(t = i,2,3); ^4/ = ^4-W- (l6)

The interpretation of this simple result, and especially that of the
meaning of <£4, is left to the reader.

We shall now pass to a remarkable and instructive graphic
representation of the Lorentz transformation, due to Minkowski.*

Minkowski calls a space-point at an instant of time, i.e. the whole
tetrad of values x, y, z, t, a -world-point (Weltpunkt), and the four-
dimensional manifold of all possible systems of values x, y, z, t the
world (die Welt). Thus, a point of the world represents a material,
or, in Minkowski's terminology, a ' substantial J particle at a certain
instant. Suppose that the particle can be recognized and watched

* H. Minkowski, * Raum und Zeit,' lecture delivered during the meeting of
the * Naturforscherversammlung ' at Cologne, 1908, Physik. Zeitschrift, Vol. X.,
p. 104, 1909, reprinted, with a preface by A. Gutzmer, by B. G. Teubner,
Leipzig and Berlin, 1909.

S.R. I

130 THE THEORY OF RELATIVITY

during its whole history. Then a one-dimensional continuum, con-
tained in the four-dimensional world, may be constructed, whose
element has the components

dx, dy^ dz, dt

along the space- and time-axes, and which represents the history
of the particle. This line, whose points may be uniquely referred
to the parameter /, say, from — oo to + oo, is called a world-line
(Weltlinie). Thus the whole world would consist of a maze of such
world-lines, and the physical laws would find 'their most perfect
expression in the mutual relations obtaining between these world-
lines.' This, of course, can be only an ideal task, and
in putting it before the eyes of physicists and mathematicians,
Minkowski, no doubt, was very well aware how far we are from its
accomplishment.

If instead of a particle or substantial point we have a body of
finite space-extension, then drawing through each of its points a
world-line, we shall obtain a tubular portion of the four-dimensional
world, which may be called a world-tube. In his previous paper,
of 1907,* Minkowski calls it a space-time filament. The utility
of the conception of a space-time filament or tube in mechanical
problems and those concerned with the motion of electrons is
obvious.

The world-line of a particle will in general be curvilinear, e.g. for
any non-uniform motion, whether the particle's path or orbit in
ordinary space be curvilinear or its velocity be changing in absolute
value. But if the particle is moving uniformly, with respect to a
given system S (x, y, z, t\ then its world-line will be a straight
line, which means only that the corresponding equations obtaining
between the four variables will be linear. In particular, if the
particle is at rest in S, then its world-line will coincide with the
/-axis, this axis, as also the axes of x, y, z, being considered as
straight lines in the four-dimensional world.

The complete representation cannot of course be given, either
by a plane drawing or by a three-dimensional model. f But this is
no serious objection against Minkowski's method. For, first of all, it

* Grtindgleichungen fur die elektromagnetischen Vorgdnge, p. 47.

fFor a remarkable attempt to obtain a geometrical image of Minkowski's
world by means of systems of spheres see a paper by H. E. Timerding in
Jahresbericht der deutschen Math. Vereinigung, Vol. XXI. 1913, p. 274.

GEOMETRIC REPRESENTATION 131

is very advantageous, especially for the trained geometer of our days,
even merely to think and to speak about these relations in terms of
four-dimensional geometry. And then we can help ourselves by
taking various sections of the four-dimensional world, by constructing
three-dimensional models (x,y, /, ory, z, t, etc.) or, still better, plane
drawings in / and one of the space-axes.

It is such a graphic representation that we are offered in
Minkowski's inspired lecture.

Let B^OB (Fig. 12) be the axis of rf, and A^OA that of x*
Draw the straight line L^OL bisecting the right angle A OB. This
line would represent the world-line of a particle moving uniformly,

FIG. 12.

along the axis of x, with the velocity of light c. Now, according
to one of the assumptions of the theory of relativity, the velocity v
of any particle is always smaller than ^, or at least does not exceed c.
Consequently no world-line will be steeper than, or even as steep
as, L^OL or N^ON. Every world-line passing through O, i.e.
belonging to a particle for which x = o at the instant /=o, is entirely
confined to the region consisting of LON and L^ON^. For, to
penetrate into LON^ or NOL^ the particle would have to move,
at least during a certain part of its wandering, with a hypervelocity.

* The plotting of x and the time against one another has, of course, nothing
novel about it. It is familiar to everybody from elementary text-books on
mechanics by the name of a 'displacement curve.' But none the less its
application to relativistic connexions has been a happy idea.

i THE THEORY OF RELATIVITY

Lut OB' be a world-line representing a particle in uniform motion
with velocity v = cfi. Then

tan ^ = ft

where \j/ is the angle BOB'. Notice that our previous angle w,
endowed with the remarkable additive property with regard to the
composition of parallel velocities, is connected with this real angle
BOB' by tan w = i tan j/\ By what has been said above, the absolute
value of the trigonometric tangent of this angle is smaller than unity,

| tan ^ | < i.

Now, to obtain a representation of the Lorentz transformation
from S(x, /) to the system S'(x't t') attached to our uniformly
moving particle, draw the hyperbola

x2-W=-i (17)

and the conjugate hyperbola

x* -&*=•!, (18)

of which the previous L^OL, N^ON, given by

«?-*»*•«* (19)

is the common pair of asymptotes.

In order to represent the particle as being at rest, i.e. in order to
pass from S to ,5", take OB', instead of OB, as the new axis of
time, that is to say of ct ', and as the axis of x' a straight line OA',
such that

or

and, instead of OA and OB, the segments OA' and OB' as the
units of x and ct', as explained in Fig. 12. The obvious proof
that this is equivalent to the Lorentz transformation x = y (x - vt\
f ' = y(t-vxjc^), is left to the reader. Notice further that, by con-
struction, OA' and OB' are conjugate semi-diameters of the hyperbola
x^-c^ = - i, as were also OA and OB.

Thus, the Lorentz transformation consists in passing from one
to another pair of conjugate semi-diameters of the hyperbola
X2 _ citi =s - 1 and in taking their lengths as the new units for
x and ct.*

*Here, as before, x, that is to say x for S as well as the new x' for S', is the
coordinate measured along v, the velocity of S' with respect to S.

• GEOMETRIC REPRESENTATION 5

The hew x- and /-axes are obtained by turning each c. ^it-
old ones, towards or away from the asymptote OL, through the

angle

j/' = arc tan /?,
not exceeding 45°.

Since ^2-/r2/2 is invariant with respect to the Lorentz trans-
formation, the asymptotes L^OL and N^ON and the hyperbolae
are fixed, i.e. remain always the same no matter whether x, t or x', t'
or x", /", etc., are chosen as variables. The same property belongs
of course to the whole system of hyperbolae

x*_<*t* = _K2

and of the conjugate hyperbolae

where K is any real number. The asymptotes may be considered
as a particular, limiting case of these curves, corresponding to K = o.
The reader is recommended to compare the case under con-
sideration with that of an ordinary rotation of a plane, say x, y, in
itself, when x2+y2 = K2 is invariant, giving circles, instead of
hyperbolae, as permanent paths of the points of the plane, and
a single fixed point K = O instead of a pair of straight lines. In
connexion with this remark hyperbolic functions may conveniently
be introduced, to replace the ordinary sine and cosine. Writing

tan ^ = tanh a, (20)

Fig. 12 will easily lead to the formulae

x = x cosh a - ct sinh a \
ct' = ct cosh a - x sinh a, j

which agree with (n), since, by (20) and (12),

<t) = ia. (22)

Remember that, by the definition of the hyperbolic functions,
sinh a = - tsin(ia),
cosh « = cos (ia).
Notice that, the region of OB1 being LON* a time-axis can be

* For positive values of /, and L^ON^ for negative values of /, and similarly as
regards LON-^ when the .r-axis is in question.

134 THE THEORY OF RELATIVITY

drawn from O through any world-point situated in this region, that
is to say, through any point for which

^2-^2/2<o. (230)

Similarly, the region of OA' being LON^ an #-axis can be drawn
through any world-point for which

so that any of such world-points can be made simultaneous with O.
This is an eminently characteristic feature of the new doctrine as
distinguished from the old system of physics in which simultaneity
was an absolute property of events, independent of our choice of
a standpoint. It is plainly an immediate consequence of the reform
of the concept of simultaneity introduced by Einstein. Pairs of
events are or are not simultaneous according to the choice of our
standpoint, i.e. of one out of an infinity of legitimate systems
•S1, S', etc., in exactly the same way as pairs of space-points have
or have not equal values of x and y (or y and z, or z and x)
according to our choice of the coordinate-planes. There is thus
far an intrinsic similarity, a kind of coordinateness, between space
and time, or as the Time Traveller, in a wonderful anticipation of
Mr. Wells, puts it : ' There is no difference between Time and Space
except that oiir consciousness moves along it.1*

The process of passing the time-axis through a world-point corre-
sponding to a given particle, since it brings it to rest, is often
referred to as transforming that particle to rest.\ In view of the
above property, a vector ('world-vector,' to be treated fully further
on) in the plane #, <tf, drawn from O to any point of the region
limited by LON, or L^ON^ satisfying the condition (230), may
be called a time-like vector, and a vector drawn from O towards any

*H. G. Wells: The Time Machine, 1898 (Tauchnitz edition), p. 13. It is
interesting to remark that even the forms used by Minkowski to express these
ideas, as ' Three-dimensional geometry becoming a chapter of the four-dimensional
physics,' are anticipated in Mr. Wells' fantastic novel. Here is another sample
(loc. cit. p. 14), illustrative of what is now called a world-tube : ' For instance,
here is a portrait [or, say, a statue] of a man at eight years old, another at
fifteen, another at seventeen, another at twenty-one, and so on. All these are
evidently sections, as it were, Three-Dimensional representations of his Four-
Dimensioned being, which is a fixed and unalterable thing.' Thus, Mr. Wells
seems to perceive clearly the absoluteness, as it were, of the world-tube and the
relativity of its various sections.

t In German, ' Auf Ruhe transformieren. '

GEOMETRIC REPRESENTATION 135

point of the remaining region, LON^ or L^ON, i.e. satisfying (23^),
a space-like vector.* On the border between these two classes of
vectors we have singular vectors, drawn from the origin to any
point of the asymptotes, i.e. coinciding, in this bi-dimensional case,
in fact, with parts of the asymptotes, and characterized by x2 - W- = o.
For / > o the world-point at the end of a singular vector represents
a particle when it just receives a light-signal from O, that is to say,
a signal started at x = o at the instant / = o. Similarly, for / < o,
the end-point of a singular vector represents a particle just at the
instant when it sends a light-flash which arrives at x = o at the instant
/ = o. Or, as Minkowski puts it, L-^ON^ consists of all the world-
points that send light towards O, and LON of all those that
receive light from O.

Notice that x = +ct, if transformed, gives x = ± ct', which follows
from the invariance of x- - c^ (together with the requirement x = x,
t' = t for z> = o), and is only a verification of the assumption, made
at the outset, that the velocity of light in empty space is the same
for all legitimate systems of reference. In this case both x and /
are reduced by the Lorentz transformation in the same ratio. In
fact, substituting x = ct in x' = y(x — vt\ t' = -y(t — vx\c<1\ we obtain

Thus far we have considered besides / one independent variable
only, the space coordinate x. Accordingly, any world-line, traced in
that bi-dimensional diagram, has been the representation of a particle,
or, in the limiting case, of a flash of light travelling along a straight
line, the X-SLXIS. Now, bring in the coordinate y. Then the resulting
three-dimensional diagram or model will be appropriate to represent
the motion of a particle, or the propagation of light, in a plane, the
plane of x, y. Return to Fig. 1 2, and imagine the axis of y to be
drawn through O perpendicularly to the paper. To obtain the
required representation, we have only to spin the two hyperbolae
of Fig. 12 and their asymptotes round B^OB as axis. The two
branches of the hyperbola (17) will generate a hyperboloid of revolu-
tion of two sheets

c*t*=-i, (24)

and the two branches of the hyperbola (18), exchanging roles after

* If we are to translate thus the names used by Minkowski : zeitartiger and
rauinartiger Vektor respectively.

136 THE THEORY OF RELATIVITY

a rotation through 180°, will give rise to a hyperboloid of revolution
of one sheet

f2/2=i, (25)

which will be cut by the jy-axis in a pair of points, say, C and Cl ,
one above and the other below the paper, while the asymptotic lines
will generate" a right cone

*2+j2 + s2-^2/2 = o, (26)

the asymptotic cone of the hyperboloids. As regards this conic
surface, let us distinguish its two parts L^ON^ and LON (revolved),
corresponding to negative and positive times respectively, and let
us call the first the fore-cone and the second the aft-cone of O*
The fore-cone consists of all world-points, out of those under con-
sideration, which 'send light' towards O, and the aft-cone of all
those which 'receive light' from O. Any vector drawn from O to
a world-point contained within the fore- or aft-cone will be a time-
like vector, and vectors drawn from O to any point of the remaining
region of the world, outside the cones, will be space-like vectors.
Now, let v be the ordinary vector-velocity of a particle in uniform
motion, and let it have any direction whatever in the plane of x, y.
Then the world-line of this particle will be a straight line passing
through O in the plane v, OB, and including with OB, the original

time-axis, the angle

^ = arc tan ft

where f$ = vjc. To transform the particle to rest, take this world-
line as the axis of ct', and to obtain at the same time the new
coordinates X ', y turn the old plane xy through the angle ^ round
an axis passing through O and perpendicular to both v and OB.
For the moment, call the coordinates measured in the :rjF-plane,
along v and perpendicularly to it, f and ^ respectively. Then the
turning round of that plane from its original position (/=o) will
amount to writing

<r/=£.tan^«/J£

On the other hand, we have

*+j*-£+V

for any point of the plane xy, so that (25) will become

f + J?-C*t*=l.

* Minkowski, ' VorkegeV and ' Nachkegel. '

GEOMETRIC REPRESENTATION 137

The intersection of the new plane, x'y, with the surface (25), will,
therefore, be given by

Now, /32<i. Thus the x'y plane will cut the one-sheeted hyper-
boloid in an ellipse. To complete the Lorentz transformation we
have only to take the semi-diameters of this ellipse as Me new units
of length measured from the origin along any direction in the x'y
plane. The major principal axis of this metric ellipse will be
contained in the plane v, OB, and the other axis will be normal
to it. This ellipse of our graphical representation will, of course,
in the new units of length, be a circle, i.e. x'2+y'2 = i. So also did
the old plane of coordinates (xy) cut the one-sheeted hyperboloid in
a circle x2 +y2 = i. This is seen at once to agree with the in variance
of x2 + y2 — t2t2. We have generally

and since the sections under consideration are obtained by putting
/ = o, /' = o respectively, the

x2 +y2 — i

has for its ^'-correspondent the circle

x"2+y'2= i.

The neiv unit of time, i.e. of ct' , is again represented by the segment
of the r/'-axis cut off by one of the sheets of the two-sheeted
hyperboloid of revolution, i.e. by the semi-diameter conjugate to
the plane x'y. So also was the old time-axis, OB, conjugate to
the old plane of coordinates (xy), and the unit of ct was the semi-
diameter OB.

To resume this three-dimensional graphic representation:

The Lorentz transformation consists in passing from one to
another set composed of a time-like semi-diameter and the
conjugate space-like semi-diameters of the hyperboloid
tf+f-W^ _x>

and in taking the lengths of the new semi-diameters as the units
for the time (ct') and for the space-coordinates ; the units of
length being thus given in each case by the semi-diameters of the
ellipse cut out from the one-sheeted hyperboloid x2 +y2 - c2t2 = i
by the plane of coordinates.

138 THE THEORY OF RELATIVITY

The new time-axis and the new coordinate-plane are obtained
by turning each of the old ones, towards or away from the
asymptotic cone round an axis passing through O and perpen-
dicular both to the old time-axis and to the velocity v of the
new system with respect to the old one.

Having gone through all of this, we can now pass to the most
general, four-dimensional case. Here, it is true, our imagery fails
us. But we can still advantageously avail ourselves of the geo-
metrical language as a guide to, and as a short expression of, the
analytical process involved.

Instead of the hyperboloidic surfaces we have now the two-' sheeted '
hyperboloidic space or, as we may conveniently call it, the double
hyperboloid

rt-cW^xt+yt + zt-cW^ -i (27)

and its conjugate, the one-' sheeted ' hyperboloidic space or the single
hyperboloid

f2 - cW = x* +y* + z*- c*t* = i, (28)

with their common asymptotic conic space

r2 _ ,2/2 = ^2 +2 + Z2_ ,2/2 = Q

consisting of the fore-cone / < o and the aft-cone t > o, as before,
with the only difference that these, like the hyperboloids, are now
three-dimensional entities.

The /-axis cuts the double hyperboloid (27) in a pair of points,
namely

x=y = z = o, ct=\
and

X=y = Z = 0, tt=—I.

Take the first, contained in the positive ' sheet.' Call it P, so that
OP, a semi-diameter of the hyperboloid (27), is the old time-
axis, and the length of this semi-diameter is the unit of ct. The
space t = o, that is to say, the ordinary space-manifold x, y, z is the
three-space conjugate to the semi-diameter OP, just as the ^jF-plane,
in the previous case, was conjugate to OB (Fig. 12). Now, instead
of P9 take any other point P' of the positive sheet of (27), and
consider OP' as the new time-axis and the length of this semi-
diameter as the unit of ct' . Turn the xyz-spa.ce (/ = o) which cut
the single hyperboloid (28) in a sphere,

GEOMETRIC REPRESENTATION 139

round the plane passing through O and perpendicular to v, till this
space, or pencil of semi-diameters, becomes conjugate to the
semi-diameter OP'. Then it will become the s^Y-space. This
space cuts the single hyperboloid in an ellipsoid (ellipsoidic surface).
Take the semi-diameters of this ellipsoid as the new units of length
measured from the origin along any direction in the x'y'z -spa.ee.
Then the Lorentz transformation, from S to S', will be completed,
and the new metric surface which, from the ,S-point of view, is an
ellipsoid of revolution will for the ^'-standpoint become a sphere,

So also was the old metric surface, viewed from the old standpoint,
a sphere of unit radius. Remember that OP' is time-like, i.e.
contained within the four-dimensional region bounded by the three-
dimensional cone, but otherwise the choice of this axis as a
time-axis is free. The possible positions of P' constitute a triple
manifold, namely all the points of the positive sheet of (27). Thus,
the systems S'(x, y', z', /') equally legitimate with 6" are co3, as
has been repeatedly observed.*

To resume what has just been said with regard to the general,
four-dimensional case :

The Lorentz transformation consists in passing from one (time-
like) semi-diameter OP and the pencil of conjugate (space-like)
semi-diameters of the hyperboloid r1 - <r2/2 = - i to another semi-
diameter OP' with its corresponding pencil of conjugate semi-
diameters, and in taking the lengths of the new semi-diameters
as the units of time (cf) and of space-coordinates; the units of
length being thus given in each case by the semi-diameters of the
ellipsoid cut out from the hyperboloid r- - <r2/2 = i by the new
space of coordinates.

The property of two lines OPl and OP2 being conjugate may be
expressed analytically by the equation

where xlt ylt zlt ct^ and x2, y2, z2, <r/2 are the values of the four

* The simple turning round of x, y, z, leaving jc2 +y2 + z2 invariant, being always
left out of account. Having once assumed the isotropy of space, we have, speaking
physically, no need to consider such rotations. And with regard to their mathe-
matical r61e, see Chap. VI.

140 THE THEORY OF RELATIVITY

variables defining the world-points Pl and P^ respectively, or, using
the ordinary vectors rx and r2,

or finally, writing l=ict,

By an obvious analogy such lines OPlt OP2 are also called
mutually perpendicular or normal lines in the world x, y, z, L
Notice that this property of a pair of lines is invariant with
respect to the Lorentz transformation, i.e. that (30) is trans-
formed into

(r1'ra')+/1'/s'~0-

In other words, conjugate diameters remain conjugate, indepen-
dently of the choice of a reference-system. ' This is obvious, at least
in two and three dimensions. More generally, for any pair of lines

OPlt OP2,

(31)

as the reader himself may prove, using for instance the form (ib)
of the Lorentz transformation, and noticing that

(€r] -€r2) ~ 72^~2(riv)(r2v) = (rir2)

identically. Thus, the invariance of orthogonality is but a particular
case of the invariance of

We shall return to the last property later on.

Given the origin O (and any world-point can be made the
origin), the set of any four values of

x, y, ^ /,
or, more generally, of any four scalar magnitudes

which are transformed like x, y, z, / respectively, and of which the
first three are real and the fourth purely imaginary, defines what is
called a world-vector or space-time vector of the first kind* (Minkowski)
or a four-vector (Sommerfeld).

* To' be distinguished, later on, from those 'of the second kind' or 'six-vectors/

GEOMETRIC REPRESENTATION 141

Thus, if (30) is satisfied, the four-vectors OP1 and OP2 are said
to be perpendicular to one another. Generally, if

(W1W2)+^2 = 0> (32)

then Wj , j^ and W2, s^ form a pair of perpendicular four-vectors. Here
Wj is the ordinary or three- vector whose components are wx, wy, wgt
and w2 has a similar meaning, while (WjWs) is, as before, the
ordinary scalar product of wls W2.

Any four-vector drawn from O to a world-point contained within
the asymptotic cone, i.e. such that r2 - c-fi = r^ + /2 < o or, more
generally, any four-vector w, s, such that

is called, as in the two- and three-dimensional cases, a time-like vector,
while four-vectors satisfying the condition r2 + /2>o or, generally,

w2 + s2>o, (331)

are called space-like vectors.

The reader will easily prove that if one of a pair of normal four-
vectors is time-like, the other is space-like, or that, in other words,
if one is contained within the asymptotic cone, the other is
outside it.

Again, as in the above special cases, any vector drawn from O
towards a point of the asymptotic cone, whether the fore- or aft-cone,
is called a singular four-vector. The analytical expression of a singular
vector is r2-^2/2 = /2 + /2 = o, or, generally,

W2 + j2 = 0. (34)

Finally, as in the less-dimensional cases, the aft-cone may be
said to consist of all world-points which 'receive light' from 6>,
and the fore-cone of all those that 'send light' towards O.

Fig. 13, which is Fig. 12 redrawn with the omission of the
arbitrary axes, and thus contains only what is 'absolute' or
independent of the choice of such time- and space-axes, may aid
the reader in remembering the meaning of the various names
employed in the above representation. This figure is drawn per-
spectively (in three dimensions, of course), so as to show that the
hyperboloids (27) and (28) are hyperboloids of revolution, the
former consisting of two disconnected ' sheets ' and the latter of one
* sheet.' We may mention further that the world-region contained

142

THE THEORY OF RELATIVITY

within the fore-cone (left) was called by Minkowski this side of O
and that contained within the aft-cone that side of O. Every world-
point of the first region is necessarily (independently of the selection
of a reference-system) or essentially earlier, and every world-point of
the second region is essentially later than O. Any point of the
remaining, cyclical, region of the world, called the intermediate
region, can be made simultaneous with or earlier or later than
O (i.e. can be given a value of / = or < or > o) by an appropriate
choice of the time-axis, and is therefore essentially neither earlier
nor later than O. This region is the domain of all space-like four-

ctor

FIG. 13.

vectors which can be drawn from O. Between the time-like and
space-like classes of world-vectors are the singular vectors, composing
the cones which are three-dimensional entities.

This partitioning of the world and the characteristic properties
of the cones are obviously conditioned by the assumption that nc
particle, or at least, no legitimate system, can ever move (relatively
to another one) with a velocity v exceeding that of light in empty
space. In classical physics there was no limit whatever to v. The
Newtonian transformation follows from the Lorentz transformation
by taking oo instead of c, or, figuratively, by widening both the
cones till they coalesce with one another in a plane, squeezing out
the space-like four-vectors and opening the whole world to the time-
like vectors. Any straight line would, in the Newtonian world,

THE MATRIX METHOD 143

represent a possible uniform motion of a particle with respect to
certain frames of reference.

So much as regards the geometric representation of the Lorentz
transformation.

Now for its analytical expression and the methods of dealing with
the world-vectors.

Minkowski, though availing himself now and then of the four-
dimensional vector language and ideology, made a systematical
and extensive use of Cay ley's calculus of matrices.* Thus, the
fundamental world-vector r, / and, more generally, any space-time
vector of the first kind w, s is considered as a matrix of i row and
4 columns, say,

X=\x,y, z,l\ (35)

and, in general,

W= wx, wy, wz, s |.

The transformed world-vector r', /' will then be another matrix
of i x 4 constituents,

X' = \ *',/, *', /' |, (35')

which is obtained from X by taking its 'product' into a certain
matrix of 4 x 4 constituents,

all> a!2> a!3» a!4

a21» a22> a23> a24

a31> a32> a33> a34
a,

(36)

*44

and which is written simply

X' = XA. (37)

Thus, the Lorentz transformation is expressed by the matrix A
taken as a postfactor of the world-vector to be transformed. This
matrix is characterized by the condition that its determinant is + i,

i, (38)

and further that all of its constituents containing the index 4 once

*Cf. Minkowski's Grundgleichungen, already quoted, §§ II et seq. Those
readers who are not familiar with this branch of mathematics may consult the Note
at the end of this chapter, where the definition of different kinds of matrices and
some rules of operating with them are given.

144

THE THEORY OF RELATIVITY

only are purely imaginary, while the remaining seven constituents
are real and the right lowermost positive :

an, a12, ... a33 real

«14> a24> a34 I

\ purely imaginary

a41> tt42> tt43 ^

The inverse transformation is represented by the reciprocal of At
which is at the same time the transposed of A, A~l = A, so that

AA = AA=i. (39)

It is this property that insures the invariance of r2 + /2. Using A
and X, we may write also, instead of (37),

X' = AX.
The short formula (37) replaces

X = an x + «21 y + agl z + <*41 /,

and three similar equations, with 2, 3, 4 as second indices. If, in
particular, the xt y, z-axes are taken along v and normal to it, and
if x', y, z are, as before, measured along the same directions, then,
as we saw,

Hence, for this particular choice of coordinate-axes the matrix
representing the Lorentz transformation reduces to

7, o, o, - ifiy

o, i, o, o

o, o, i, o

i/ty, o, o, y

(40)

The transposed matrix A which represents the inverse Lorentz
transformation is obtained from this by simply changing the sign
of /?, as it should be.

THE MATRIX METHOD 145

Writing, instead of x, . . . /, the differentiators 3/3#, . . . 3/37, we
obtain a matrix of i x 4 constituents, which Minkowski called lor,
in honour of Lorentz,

3333

lor =

"dx "dy 'dz 3/

(40

This is the matrix-equivalent of our quaternionic differential operator
Z>, as defined by (13), Chap. II. It can be easily verified that
d/ctor, ... 3/3/ are transformed in exactly the same way as x, y, z, /
respectively.* Thus, lor is covariant, or equally transformed, with
the matrix X representing the standard world-vector, i.e.

lor' = lor A. (42)

Moreover, it has the same structure as X, its first three constituents
(differentiators) being real and the fourth, 3/3/, purely imaginary.
Thus, lor, though an operator, behaves in every respect like a
space-time vector of the first kind.

We cannot stop here to consider the matrix form of space-time
vectors of the second kind and their analytical connexion with those
of the first kind (although it could be done in a few lines), for the
reader does not yet know their relativistic physical significance.
Moreover, it is not our purpose to develop fully the matrix' method of
treating relativistic questions, since we shall avail ourselves chiefly of
other methods. But one simple property of products of fF-matrices
in connexion with the above remarks is worth mentioning here,
namely that, if Wlt W% are matrices representing a pair of vectors
of the first kind (w15 s1'} W2, s2), the product

^1^2 = (WlW2) + ^2 (43)

is an invariant. For by (39), and by the associative property of
products of matrices,

* Thus, for instance, measuring x along v,
we have

whence x = y(x' -t£/')» etc., and

3 /3 3\ 3 3 3 3 3 /3
3?

S.R.

146 THE THEORY OF RELATIVITY

Thus, the orthogonality of two four-vectors, which is an invariant
property, is expressed by

fFifF2 = o.
Similarly,

lor' ZF' = lor W,
or lor W is a relativistic invariant. Notice that, similarly to (43),

"dWf l5Wu *dWz ?>S

lor W=-~r- + -~r- + -~r + ~rj '

ox oy oz ol

or, using div in its ordinary sense,

lor JP=divw + ^- (44)

So much as regards Minkowski's matrix-form of the fundamental
relativistic connexions.

Sommerfeld, whose aim was to elucidate Minkowski's ideas, re-
placed his language of matrices by a four-dimensional vector-algebra
(and -analysis) which he developed in two very lucid papers,*
and which is an obvious generalization of the familiar three-
dimensional calculus of vectors. Sommerfeld begins by drawing
our attention to the well-known circumstance that in space of three
dimensions there are two kinds of vectors to be distinguished, e.g.
vectors of the * first kind ' or polar, and those of the ' second kind '
or axial vectors. A vector of the first kind, such as a translation
velocity, is a segment of a straight line having a certain direction
(and sense) ; its components are the projections upon the coordinate-
axes. On the other hand, a vector of the second kind, such as
angular velocity, is represented by a plane surface of a certain area
with a given sense of circulation round its circumference, and its
components are the projections of that area upon the coordinate-
planes. Consequently, the components of a vector of the first kind
should be written with single indices, vx, vy, vz, or z>15 #2, #3, while
those of a vector of the second kind, as, for instance, rotational
velocity w, with double indices, ojr,, w^, w^, or w23, o>31, w12. This
discrimination, which in three dimensions is not very important (or
at least ceases to be so when, instead of the plane area, a repre-
sentative line-segment normal to it is introduced), becomes in

*A. Sommerfeld, 'Zur Relativitatstheorie. I. Vierdimensionale Vektoralgebra/
Ann. der Physik, Vol. XXXII., 1910, p. 749, and 'II. Vierdimensionale Vektor-
analysis,' Ann. der Physik, Vol. XXXIII. , 1910, p. 649.

FOUR-DIMENSIONAL VECTORS 147

Minkowski's four-dimensional world quite essential. For here —
argues Sommerfeld — we have

(\)=four coordinate axes,

(y) = 5>z.T coordinate planes,

vz, z#, #jr, #/, j'/, z/,
and

(J) =./&«*• coordinate spaces,

#)>£, JZ/, Z#/, #_}'/.

Accordingly, we have to distinguish in the * world ' between

vectors of the first kind having four components, or four-vectors ;
those of tlie second kind having six components, or six-vectors ;

and, finally,

those of the third kind, which again have four components, and
can be replaced by their 'supplements,' which are vectors
of the first kind.

Consequently, vectors of both the first and the third kind are
called by Sommerfeld, summarily, four-vectors.

This classification will be found useful for what is to occupy us
later on. But meanwhile we are concerned only with space-time-
vectors of the first kind, which we shall simply call four-vectors.

The standard or typical example of such vectors is that drawn
from the origin O to any world-point. Call it P.* Then its
components would be, according to Sommerf eld's general notation,

These, of which the first three are real and the last imaginary, are
simply the previous

x, }', z, /.

* Sommerfeld does not use any special type of print for his four-vectors, to
distinguish them from six-vectors. A certain uniformity of notation was
introduced later by Laue, he. cit. But we shall not want very much of it for
our subsequent purposes.

148 THE THEORY OF RELATIVITY

What Sommerfeld denotes by \P\ and calls the size of the vector
P, or its length, i.e. the * length ' of the corresponding four-dimensional
straight line, is the positive (or positive imaginary) value of

v/*2 + / + z2 + /2 =

or of vV- - <r2/2.

The length of this, and of every other, four-vector is invariant
with respect to any Lorentz transformation. It is its only invariant.

Notice that the length, thus denned, of a four-vector may be
either real, or purely imaginary, or nil, according as we have what
was previously called a space-like, a time-like, or a singular vector.

If A, B be any pair of four-vectors, the sum of the products of
their corresponding components is called their scalar product, and
is denoted by (AB). Thus

(AS) = (AXBX + AyBy + AZBZ + AlBl\ (45)

Guided by the analogy of ordinary vector-algebra, Sommerfeld
defines then the direction-cosine of A relative to B, or vice versa^
by writing

(AB} = A\.\JB\.<xx(A,£). (45*)

Consequently, when

the four-vectors A, B are said to be perpendicular to one another
This is identical with the previous definition of pairs of perpen-
dicular vectors.

What Sommerfeld calls the ' vector product ' of A, B cannot here
occupy our attention. For such a product is a (special) six-vector,
which as yet is unfamiliar to us.

As to the Lorentz transformation itself, it appears in Sommerfeld's
treatment as a rotation of the system of four axes. Let P be any
four-vector, and JPX, etc., its components along the old axes ; then
Sommerfeld defines the components of P along the new axes by

Px, = Px cos (#', x) + Py cos (#', y) + Pz cos (*', z) + Pl cos (*', /), (46)

and by similar formulae for Py, P^, Pv. Here, the meanings of
the cosines are defined by (45^). If the x'-axis is space-like, then
the first three cosines in (46) are real, while cos (x, /) is purely
imaginary, like JPi9 so that P^ is real. Similarly, the y'- and z'-axes
being space-like, JPy, Pz~ will be real. And the /'-axis being time-

FOUR-DIMENSIONAL VECTORS 149

like, Pf will be purely imaginary. In order to show that the
projection- or component-formulae (46), etc., are identical with those
of the Lorentz transformation, Sommerfeld considers the particular
case of rotation round the jz-plane, i.e. in the #/-plane, when

cos (x't x) = cos (/', /), say = cos w,

cos (#', /) = - cos (/', x) = \/i - cos2w = sin w,

cos (/, y} = cos (z', z) = i,

while all other cosines vanish. Here we have, obviously, coso»i,
so that the angle o>, as well as its sine and tangent, are purely
imaginary, and the absolute value of the latter is < i. Consequently
we can write

tan <o = i /?, cos o> = ( i - /?2)~T = y, sin w = i /3y,

so that (46), etc., are at once reduced to the formulae (n), on p. 127,
with the same meaning of <o, provided that the new system of space-
axes (x'y'z) moves relatively to the old one (xyz) with the uniform
velocity v = cfi along the X-SLXIS. There is in fact no difference what-
ever between Sommerfeld's and Minkowski's method of representing
the relativistic transformation.

It is true that the systematic use of the four-dimensional vector
language may offer some advantages, when compared with that
based on the use of matrices. But, on the other hand, there are
rather important arguments which may be brought forward in defence
of the matrix-method. Thus, for instance, Sommerfeld's ' scalar
product,' say (AB), is the same thing as Minkowski's product of the
corresponding matrices, W^ W^ (43). But whereas the in variance
of Wl W^ is seen at a glance, viz. by writing, in virtue of the
fundamental formula (39),

the invariance of (AB) cannot be proved without splitting the four-
vectors into their components and multiplying out expressions like
(46) and adding them up. For Sommerfeld's only definition of
' scalar product ' (45) is of such a character. It is essentially
Cartesian, not vectorial. Of course, we know that, in three-
dimensional space, the scalar product of a pair of vectors can be,
and generally is, defined without any reference to axes, so that its
invariance with respect to space rotations requires no proof. But

150 THE THEORY OF RELATIVITY

this does not by itself enable us to see the invariance of (AB], when
we are asked to pass into the four-dimensional world, where our
imagery fails us. Similar remarks could be made with respect to
other points of Sommerfeld's method of treatment. But discussions
of this kind need not detain us here any further.*

In the sequel we shall not avoid either of these two methods of
analytic expression. In fact, we shall now and then profitably
employ matrices as well as world-vectors. But principally we shall
avail ourselves of the language of Hamiltoris quaternions, the
utility of which for relativistic purposes I have endeavoured to show
in two papers.! I may notice that Minkowski himself (Grundgkich-
ungen, p. 28, footnote) despised Hamilton's calculus of quaternions
as ' too narrow and clumsy for the purpose ' in question. But,
notwithstanding that, I am still under the impression that quaternions
are admirably suitable for most, if not for all, relativistic needs.
We had a sample of the conciseness of Hamilton's language in
Chapter II., when we saw how easily the four vector equations
of the electron theory are condensed into a single quaternionic
equation, viz. Z>B = C. But in advocating here the cause of
quaternions I am doing so not only because they furnish us very
short formulae and simplify their handling. Quite independently
of this, the quaternion seems to me intrinsically better adapted
than the world-vector to express that * union ' of time and space
which was (too strongly, perhaps) emphasized by Minkowski. For,
although there is a certain union between the two, which manifests
itself when we pass from one system to another, there is no total
fusion. In each system, out of the four scalars x, y, z, /, the first
three are more intimately bound to one another than any of them
to the last one. The first three are artificial components of a vector,
r, which certainly is a more immediate entity than each of them.
Now, in a four-vector, as well as in a matrix, x, y, z, / are, as it

*Nor can we enter here upon a paper of E. B. Wilson and G. N. Lewis,
Proc. Amer. Acad. of Arts and Sciences, Vol. XLVIII., Nov. 1912, p. 389, in
which an attempt is made to work out the four-dimensional vector-algebra and
-analysis, ab ovo, starting from a number of quasi-geometric postulates.

^tPhil. Mag., Vol. XXIII., 1912, p. 790, and Vol. XXV., 1913, p. 135; alsc
Bull, of the Societas Scientiarwn Varsaviensis, Vol. IV. fasc. 9, communicated
in November, 1911. I wish to mention here that Dr. G. F. C. Searle has
drawn my attention to a paper of Prof. Con way, Proc. Irish Acad., Vol.
XXIX. Section A, March 1911, in which some of my results are arrived at.
Particulars of comparison are left to the reader.

QUATERNIONIC METHOD 151

were, on entirely equal footing with one another, being the four
' components ' of the former, or the four c constituents ' of the latter.*
On the other hand, a quaternion q has a distinct vector part, V. q or
simply V^, and a scalar part, S^, and none of the components of
the former can be confounded with the latter. Now, the position of
a particle is determined by a vector (in its ordinary sense), and its
date by a scalar. What then more natural than to take the first as
the V and to embody the second in the S of a quaternion? We
could insist upon loosely juxtaposing both entities, and write simply

r, /.

But, if instead of the comma the plus sign is used, we have just
enough of ' union ' to express the relativistic standpoint, and yet
enough distinction not to amalgamate time and space entirely.

Let us therefore combine the position vector r of a particle with
its date, /=*<:/, into a quaternion,

? = /+r, (47)

which, if it needed a name of its own, we might call the position-
quaternion. Those who are particularly fascinated by the world-
concept can consider this ' position ' to be the ' position in the
world.' But, in fact, the above provisional name is simply an
abbreviation for 'position-date quaternion.'

The conjugate of q, i.e. Hamilton's K^, will be denoted by qc.
Thus,

The reader must not be afraid of quaternions. If he is familiar
with the elements of ordinary vector-algebra, the following short
remarks will enable him to understand thoroughly all of our sub-
sequent calculations.

1. Without returning to Hamilton's original expression of a quaternion
as the ' ratio ' or the quotient of two vectors, he can conveniently define
it from the outset as the sum of a scalar and a vector^ using for the latter
heavy type. Thus

will be a quaternion, whose scalar part is <r, and whose vector part is A,

* It is true, that the fourth, /, is imaginary, while the first three are real, but
this does not seem to me to emphasize the distinction sufficiently.

152 THE THEORY OF RELATIVITY

2. The conjugate ae of the quaternion a is defined, as above, by

#c = cr-A,
i.e. by ac = Sa- Va.

3. Two quaternions «, b are said to be equal if both their scalars and
their vectors are equal to one another. Thus,

a = b
means the same thing as

Va = Vt> and Stf = S£.

4. Quaternions are added to one another by adding separately their
scalars and their vectors. Thus

means the same thing as

Now, since the addition of scalars and the addition of vectors are both
commutative, the commutative property belongs also to the sum of
quaternions,

And for the same reason the associative law holds for the sum of any
number of quaternions. Thus

a +
so that both sides may be simply written

5. Subtraction of quaternions, and the change of the sign of a
quaternion are at once reduced to the same operations applied to scalars
and vectors. Thus, if «=

— a= — cr — A.
Also, by 4, a - b = - b + a.

6. Two quaternions, a = cr + A and £ = r + B, are multiplied by the
formula

ab = err + rA + crB + AB,

where the first three terms require no further explanation, and the last is
defined to be a quaternion

AB=VAB+SAB,

such that VAB is identical with the 'vector product' and SAB is the
negative ' scalar product,' both supposed to be known from ordinary
vector algebra. Thus, in our usual notation,

AB=VAB-(AB).

ELEMENTS OF QUATERNIONS 153

The minus sign is introduced to suit the whole of Hamilton's calculus ;
I do not think there is any trouble in doing so. Ultimately, the product
ab of a pair of quaternions is given by

Vab = rA + o-B + VAB.

Thus, ab, and similarly, the product of any number of quaternions, is
again a quaternion, with uniquely determinate vector and scalar parts.

Both (AB) and VAB being distributive, quaternion multiplication is
distributive, i.e.

a[b+c]=ab + ac,

\b-\-c\a-ba-\-ca.

It can easily be shown that it is also associative, i.e. that
a . bc=ab . c,

so that both sides may be simply written abc. The same thing is true
of the product of any number of quaternions. It is chiefly this associative
property which makes Hamilton's calculus so powerful.
From the above formulae we see that

because (AB), like <rr, is commutative. On the other hand we have,
generally,

Vba ± Vab,
because VBA =- VAB.

Thus, multiplication of quaternions is, generally, not commutative,

ba =/= ab.

It becomes commutative only when VAB vanishes, i.e. when A||B,
or V<z || Vb. This is, for instance, the case for a pair of conjugate
quaternions, and, consequently, we have, for any quaternion a,

7. Writing again <7 = o- + A, we have, by 6,

"4

aae = aca = <r2 + A*,

where A2 = (AA). Thus aac is always a pure scalar. Its square root is
called the tensor of the quaternion a, and is denoted by T<z,

If it is real, the positive value of the root is taken, and if purely
imaginary, the positive imaginary value of the root is taken. (In cases
of complex values, when ambiguity of T might arise, special explanations

154 THE THEORY OF RELATIVITY

will be given.) But the chief thing is to keep in mind the formula for
the square of the tensor,

(Ta)2 = aac = aca,

which is called the norm of the quaternion a.

Let a be the unit of A, so that (aa)= i and A=A&. (There is, I hope,
no danger of confounding the quaternion a with the absolute value of a,
which is i.) Then the quaternion a can be written

a = Ta . [cos a + a sin a],

or, by an obvious analogy,

tf«*T« .<«»,*

where e is the basis of natural logarithms. The factor of la, which is a
quaternion of unit tensor or a unit quaternion, is called the versor of the
quaternion a, and is denoted by I] a, so that a = Ta . I] a. The unit vector
a is called the axis of the quaternion a, and the angle a, which can be
real or imaginary, is called the angle of the quaternion a.

Thus, conjugate quaternions may be described as quaternions having
equal tensors and equal angles, but opposite axes.

Notice that if (as in the case of our above g) <r is imaginary and A real,
or vice versa, the tensor of a may vanish, though a is not simply 'zero,'
but a definite quaternion having a certain axis and a certain angle. Such
a quaternion was called by Hamilton a mtllifier, and by Cayley a nullitat.
In our physical applications we shall not avail ourselves of either of these
names, but shall adopt for such quaternions the name singular, already
used for the corresponding world-vectors.

8. The following rule, which will be often required, can easily be
proved :

The conjugate of the product of any number of quaternions is the
product of their conjugates in the reversed order.
Thus, if

m^ab,
then

9. Finally, as regards division by quaternions, it may be entirely
reduced to multiplication by what are called their reciprocals.

The reciprocal of a quaternion a is again a certain quaternion, which is
denoted by a~l and which is defined by the equation

* But this analogy cannot be pushed so far as to write in the expression of a
product of two quaternions a, b

and to invert the order of addends in the exponent. For, unless a||b, the
product ab is not commutative.

ELEMENTS OF QUATERNIONS 155

Multiply both sides by ae as a postfactor. Then ci~1aac=aei and, by 7,

Thus, the reciprocal of a quaternion is its conjugate divided by its norm.
In other words, the reciprocal of a has the reciprocal tensor, the opposite
axis and the same angle as a.
Consequently, we can also write

aa~1=i.
Thus, if we have an equation

am = b

and we wish to isolate m, we have only to multiply both sides by a~l as
prefactor, obtaining

m=a~lb.
Similarly, if

na = b,
we shall have

n = ba~l.

Notice, in particular, that the reciprocal of a unit quaternion is at the
same time its conjugate.

10. The differentiation of quaternions, with respect to time or position
in space, does not require any explanations. The definition of ' curl ' and
' div ' being supposed known from usual vector-analysis, it will be enough
to remember here what was said already in Chap. II., namely that v, the
vector part of our Z>, when applied to a vector A, gives

VA = VVA + s VA = VVA - (VA)

(as in 4, because v apart from its differentiating properties is to be treated
as an ordinary vector), or ultimately

VA = curl A -div A.

For all of our purposes we shall hardly want more than is given
in the above ten sections, — which will in the sequel be shortly referred
to as ' Quat. 1, 2, etc.'

Returning to our position-quaternion q, let us write its 5"-corre-
spondent, or the transformed quaternion

?' = /' + r'. (47')

Since l=ict, i.e. /= -*//<", we have, by (i^), p. 124, and denoting
now the unit of v by u,

(48)

156 THE THEORY OF RELATIVITY

Here, it will be remembered, e is the longitudinal stretcher, whose
developed form is, by (2),

€=i+(y-i)u(u .

Now, such being the scalar and the vector parts of q in terms
of those of qy we can easily find a quaternion Q such that

?' = <2?<2- * (49)

First of all, since we know that /2 + r2 is an invariant, or that
T/ = T^, we can at once take for Q a unit quaternion,

Q = cos 6 + a . sin 0.

Thus, we have only to find the angle and the axis of Q in terms
of ft and u. Now, developing the triple product in (49), we obtain
easily, by Quat. 6,

r' = VQ?Q = i - 2 sin26> .a(ar) + sin (26"). /a,
whence, comparing with (48),

a = u; cos(20) = y; sin(20) = //ty,

and i - 2 sin2#.a(a = e, i.e. 2sin'20=i-y, which is identical with
the third of the above equations, and this, again, says the same
thing as the second. Thus, all conditions are satisfied at once, and
we have ultimately

a = u and 6 = J arc tan (i/3) = Jo>,

where w is the (imaginary) angle of rotation, as previously defined.
[Cf. (10), p. 127.] To resume:

The position-quaternion q is transformed by the operator

Ql ]Q,

the vacant place being destined for the operand.

The axis of the unit quaternion Q is u, the unit of v, and its
angle is half that of Minkowskts imaginary angle of rotation, i.e.

~ to . w £u

<2 = cos- + u.sm- = ^ . (50)

*As regards the reason why particularly this form, involving a quatefnionic
prefactor and postfactor, is sought for, see my paper in Phil. Mag., Vol. XXIII. ,
quoted before, where I gave references going back to Cayley's original discovery
(1854).

QUATERNIONIC METHOD 157

Another form of this quaternion is

Observe that, y being > i, the vector part of Q is imaginary, while
its scalar part is real.

Since Q is a unit quaternion, we have Q~l=Qa or

a property which we shall constantly use. Thus, to obtain from
(49) the inverse transformation, multiply both sides by Qc as a
post- and a prefactor. Then the result will be

q=QdQ» (49«)

as it should be, since Qc is obtained from Q by a reversal of u or v.
Again, to see once more, or to verify, the invariance of

(51)

take the conjugate of (49), which, by Quat. 8, is

?'c=<2c?c(?c.
Now, by the same formula (49), and by the associative law,

But, since qqc is a scalar, it may be written before the Q, or if you
wish, after the Qc, so that

Q.E.D.

We shall see later on, when we come to consider products of
two, or more, of such quaternions, that they are transformed with
equal ease.

Consecutive transformations assume the following simple form.
Let V1 = ^1u1 be the velocity of £' relative to 5, and V2 = z>2u2 the
velocity of S" relative to 5', and let <215 Q<> be the corresponding
transforming quaternions, i.e.

158 THE THEORY OF RELATIVITY

Then

/--Cifft

and

?'' = <22/<22 = <22<2i?<2i<22,

so that the compound transformer is

C,Ci[ ]QlQt.

In general, for non-parallel axes Uj, U2,

so that the compound transformer has not the form Q [ ] Q. This
is but the quaternionic expression of the fact, to be considered fully
in the following chapter, that a pair of consecutive three-parametric
Lorentz transformations, (48), does not generally give again such a
transformation, but is equivalent to (48) combined with a pure
rotation in ordinary three-dimensional space. In other words, the
transformations (48) do not constitute a group. But, as we saw
before, they contain sub-groups, namely for parallel velocities. Then,
and only then, Q.2Qi becomes equal to <2i<22> and the compound
transformer assumes the form Q [ ] Q. Suppose, for instance, that
the velocities vx, v2, being parallel, are also concurrent with one
another, i.e. that

U1=U, = U.

Then

so that the previous formula for the composition of parallel velocities,
o> = w1 + a>2, follows from the quaternionic form immediately.

Imitating the name 'world-vector,' we could now call ^, or qc,
the standard of world-quaternions. But the more modest name
physical quaternion will do as well. Also, to begin with, no further
specification of the ' kind ' is needed. But it may be convenient to
have a pair of short symbols, in order to compare any quaternions
with respect to their relativistic behaviour. By writing

X~q,

we shall understand that the quaternion X is covariant or, equally
transformed, with q, i.e. that

X'=QXQ

PHYSICAL QUATERNIONS 159

without taking into account the structure of X. And if X has also
the structure of q, that is to say, if it has a purely imaginary scalar
and a real vector,* then we shall write

X~q.

The latter will then be equivalent to saying that X is a physical
quaternion, viz. covariant with q. This being the case, the conjugate
of X will, of course, be also a physical quaternion, e.g.

Xc~qc.

The same notation we shall extend to quaternionic operators.
Thus, as we saw, 3/3/ and V, the scalar and the vector parts of
the operator Z>, are transformed like /, r, the scalar and the vector
parts of the position-quaternion, i.e.

D'=QDQ, (52)

and similarly, Z>c = 9/c)/ — V being the conjugate operator,

($2f)

But D has also the same structure as q. Consequently, apart from
its differentiating properties, D behaves as a genuine physical
quaternion, or

D~q.

Analogously to Minkowski's classification of four-vectors, we may
call any physical quaternion X a space-like, or a time-like, or finally
a singular quaternion, according as its norm, (TX)2 — XXC, is positive,
or negative, or zero.

But it does not seem desirable to dwell any longer upon the
formal side of the subject until our stock of materials has
been somewhat enlarged. For as yet we have only one physical
quaternion, namely q.

* If the reverse is the case, then iX will have the structure of q.

i6o

THE THEORY OF RELATIVITY

NOTE TO CHAPTER V.

(To page 143.) A matrix is any rectangular array of magnitudes or,
more generally, of symbols either of magnitude or of operation, each of
which has its assigned place, i.e. belongs to a given row and a given
column. Thus

A =

#11) #12) ...#ln
#21) #22) ...#2n

#ml) #m,2) ...#»m

is a matrix of ;« rows and ;/ columns. The first index of any constituent
aiK denotes the row, and the second the column to which it belongs.

The matrix whose rows are the columns of A is called the transposed
of A, and is denoted by A. Thus, A being as above,

A =

#11) #21) ... #/wl
#12) #22) «..#»i2
#1«) #2w) ... #»m

To specify the number of rows and columns of a matrix we may con-
veniently attach to its symbol a pair of indices. Thus, A will be Amn,
and similarly A—Anm- Or we may say, equivalently, that A is a matrix
of m x n constituents, and A a matrix of n x m constituents.

If we have a pair of matrices A = Amn and B-=Bmnt then the matrix
C—Cmnt whose constituents are sums of the corresponding constituents
of A, B (i.e. <TUC = #IK + ^IK), is written

If, in particular, B=A, the result of addition is written 2^4, and so on.
Generally, if a be any number (or symbol of operation) and A any
matrix, then the matrix C, whose constituents are £uc = cuztK) is called
the product of a into A, and is denoted by a. A.

If the matrix B has as many rows as A has columns^ i.e. if

A=At

B = B«

{where p may be equal to or different from m\ then the matrix C, of
which any constituent CM is equal to the sum of the products of the
constituents of the ith row of A into those of the /cth column of B, is
called the product of A into B, and is written

MATRICES

Thus, if A is as above, and if

^11 1 ^12, .--b\p

161

then

where

C — Cmn — A I

^22, -.

c\\ — a\\ b\\ + #12 bi\ + . . . + a\n bn\ ,

and so on, generally

. = fia b\*

Notice that, if / =/= ;«, the expression ^^4 would be meaningless. But,
since B=Bpn and A—Anm^ we can have the product /M, which will be
a matrix of^>x;« constituents. This, as can easily be seen, will be the
transposed of AB, i.e.

AB=BA.

Compare this property with Quat. 8. p. 154.

Since AB = Cmp^ it can be multiplied into a third matrix D = Dm, thus
giving rise to AB . D, which will be a matrix of nt x q constituents. It
can be proved that for such products the associative laiv holds (supposing,
of course, that the constituents themselves, which generally can be
operators, obey this law), i.e.

AB.D = A.BD. ••

Hence, both sides may be simply written ABD. The same property
belongs to the product of any number of matrices. Thus

will be a definite matrix of m x z constituents, independent of the grouping
of the factors. Notice the analogy with quaternionic products.

Let each of the constituents of the principal diagonal (from left upper-
most to right lowermost) of a square matrix U= Unn be equal i, i.e.

and let each of its remaining constituents
any matrix of n rows,

UM=M.

S.R. L

be zero. Then, if M be

162 THE THEORY OF RELATIVITY

In view of this property, U is called a unit-matrix, and may be simply
denoted by i.

Now, let M be any square matrix. Then the determinant formed of
its constituents is called the determinant of M, and is shortly written
det M. Suppose that det M does not vanish. Then there exists a definite
matrix which, multiplied into M, gives a unit matrix or simply 'unity.'
This matrix is called the reciprocal of M, and is denoted by M~l. The
above definition is written shortly

where i stands for Unn. The reciprocal is, of course, as M itself, a
square matrix of n x n constituents.

Other particulars concerning matrices will be given incidentally, as
the need arises in the subject under consideration.

CHAPTER VI.

COMPOSITION OF VELOCITIES AND THE LORENTZ
GROUP.

CONSIDER a particle moving about in an arbitrary manner in the
system S', which in its turn moves with uniform velocity v relatively
to the system S. Let p' be the instantaneous velocity of the particle
from the point of view of the ^"-observers, i.e. let at the instant /'

dr

dt' = *'

What is the velocity p of this particle from the ^"-standpoint, at the
instant / corresponding to /' ?

To answer this simple but very fundamental question of relativistic
kinematics, use the form (i<£), Chap. V., of the Lorentz trans-
formation. Then its inverse will be, as in (i^'),

and, since dtT' = €dT and d(vr') = (vdir'),

di edi' + yvdt'

Divide the numerator and the denominator on the right by dt\
and remember the meaning of p'. Then the required velocity will
follow at once under the simple form

:. P=

164 THE THEORY OF RELATIVITY

This is the vectorial expression of Einstein's famous Addition
Theorem*

As before, y = (i -/?2)~^, fi = v/c9 and e is the longitudinal stretcher
of ratio y. Thus, in Cartesians, with x measured along v, (la) will
become

*
-

But having explained this for the non-vectorial reader, we shall
henceforth use the above short formula.

By writing p', p we wished to emphasize that the latter is the
^-correspondent of the former. But we may as well look at p as
the resultant of v and p', keeping in mind that the first of these
component velocities is taken relatively to one system, S, and the
second relatively to another f system S'. Then it may be more
convenient to write for the velocities to be compounded v1? V2
(instead of v, p'), and for the resultant velocity v (instead of p).
Thus, attaching the correspondent index to y and e, we shall write

Notice that the resultant is, in general, a non-symmetrical function
of the two component velocities. It is important to know which of
these comes first, and which next. In Newtonian or classical
kinematics the resultant is simply vx plus V2 and at the same time
v2 plus vr Here the case is different. We may still speak of
' addition,' as a non-pedantic synonym of composition of velocities,
but to avoid confusion we should employ instead of the ordinary +
another symbol, say =H= , and write the above v, as given by (i),

* ' Additionstheorem der Geschwindigkeiten,' Ann. d. Phys.y Vol. XVII.,
1905; §5-

flf both were taken with respect to the same system, then their resultant
would, of course, be simply equal to their vector sum. But this is hardly
worth mentioning. For all cases of composition of velocities, which have any
physical interest, are of the type considered above, viz. imply component velocities
referred to a chain of different systems : An object /> moves in a given way
relatively to A, a third object C moves relatively to B> and so on ; find the
motion of the last relative to the first.

COMPOSITION OF VELOCITIES 165

Then the resultant of V2 and vx (i.e. the ^-velocity of a particle
moving with velocity vl relative to S', which in its turn moves with
velocity V2 relative to S) would be

where €2 is a stretcher acting along V2, of ratio y2.

In short, the relativistic composition of velocities is, generally
speaking, non-commutative.

But it is interesting and, in view of what has to come later, useful
to notice that the two vectors (i), (2), though differing in direction,
are identical in their absolute magnitude. To see this, we have
only to prove that the squares of the two vectors

and

b = v., + —
72

are equal to one another. Now, by the elementary rules of vector
algebra,

and, since €l is a symmetrical vector-operator,

(Vi-€iv2) = (eivi-v2) = 7i(viv2)-
Again, denoting by 6 the angle between va and V2,

(- fjTjY = z>22 [cos2 0 + —, • sin2 6] = z/22[i - ft2 sin2 6].
\7i ' 7i

Hence

c? = v* + v<? + 2 (v^) - ^ vfvf sin2 0 = (vx + v2)2 - ^ (Vv^g)2,

and this, being a symmetrical function of vlf V2, is at the same time
the value of b~. Q.E.D.

Thus we have the general property of relativistic composition of
velocities :

faHhTflP-fa + T,?' (3)

166 THE THEORY OF RELATIVITY

The common value of these scalars is, by (i) and by the formula
just found for a2,

K+v^-^^v^

— -- (4)

This is Einstein's famous formula for the square of the resultant
velocity, written vectorially.

Before passing to give a few examples and a certain very re-
markable geometric representation of the Addition Theorem (i),
let us approach the question of composition of velocities from
another side, e.g. by considering a pair of consecutive Lorentz
transformations.

Let again vx be the velocity of S' relative to S, but instead of
our particle take a third system S" moving relatively to »S" with the
velocity V2, the former velocity being taken from the ^-standpoint
and the latter from the ,S"-point of view, both being now uniform
translational velocities. Let ylt €1 and y2, c2 be the corresponding
meanings of 7, e. Then, /', r' being the time and the space-vector
in Sr, and /", r" the time and the space-vector in S", we shall have,
by (i£), Chap. V.,

and

r" = e2r - v2y2/' ; /" = y2 [/' - ^ far')]. (52)

Introduce the values (5^ of r' and /' into (52), and remember that,
€j being a symmetrical vector operator, (v2 . cp) = (^v.., • r)- Then
the result will be

r" = €2eir + ~ 7i7.2v2(v1r) -

(6)

The Lorentz transformations hitherto considered, of which (5^
and (52) are individual cases, involve three scalar parameters
fec> Vy> V*) or °ne vectorial parameter v. Let us therefore denote
any one of these transformations by -£(v). Thus, the two above
component transformations will be Zfo), Z(v2), and their resultant,

THE LORENTZ GROUP 167

i.e. the first followed by the second, or the transformation (6), may
be written Z^Zfo).

We know that any Z(v) leaves invariant the quadratic expression

and can therefore be considered as a rotation in the four-dimensional
world. But it is not the most general rotation, since it does not
include the rotation round the time-axis, i.e. a rotation of the space-
framework, or an equivalent rotation of the three-dimensional vectors.
If any transformation Z(v) is followed by such a rotation of r',
which does not change the value of r2 = x'2 +y- + z"2, then the
above quadratic expression will, obviously, continue to be an
invariant. Let 12 be a purely rotating operator, or what Gibbs*
called a 'versor,' i.e. such a linear vector operator that, for any
vector R,

Then the amplified or, as it is sometimes called, the general Lorentz
transformation will be given by

r' = 12[£r-vy/],

^(v, 12)

Since 12 involves three scalar data, viz. one for its angle and two
for its axis, Z(v, 12) will be a six-parametric transformation. Thus,
the above symbol Z(v) of the special Lorentz transformation stands
for Z(v, i). Notice that the scalar product of two vectors, e.g. (vr),
is not changed at all by a pure space-rotation. This is the reason
that 12 does not enter into the expression for /', and would not enter
into it even if the rotation preceded the special Lorentz transformation.

Let us now return to our Z(v2)Z(v1), as given by the formulae (6).

We have seen in the last chapter that, if the velocities vx and V2
are parallel to one another, the resultant transformation is again a
special Lorentz transformation, i.e.

where v || Vj || v.2. Now, it can easily be shown that this is the case
only for Vj || V2.

*J. Willard Gibbs, Scientific Papers, Vol. II. p. 64.

168 THE THEORY OF RELATIVITY

In fact, suppose that (6) is an Z(v), that is to say, suppose that
there is a vector v (with the corresponding y and e), such that

r" = €r-vy/"; t" = y[t — sC7*)]'

Then, remembering that this has to coincide with (6) for every r
(as well as for every /) and taking, for instance, r = o, you will
obtain, from the first of (6),

and at the same time, from the second of (6),

yv = y2[>1v2 + y1v1],
and, consequently,

7i t€2vi + 72V2] = 72 Oi

Now, this equation cannot be satisfied unless Vj and v.2 are parallel.
To see this, call lx and % the parts of vx taken along and normal
to v2, and similarly 12 and n2 the parts of V2 taken along and normal
to V15 and write V1 = l1 + n1, V2 = l2 + n2. Then, remembering that
«!, e2 are longitudinal stretchers, the above equation will assume the
form

7i [721! + ni + 72*2 + 72n2] - 72[7il2 +
or

Hence, either yi = y2=i, which corresponds to the trivial case
^ = #2 = 0, or njllng, and consequently also V1||v2. Q.E.D.

Thus, if Vj and V2 are not parallel to one another, the resultant
transformation (6) is not an Z(v). In other words, the class of oo3
transformations L(v) does not constitute a group, although it contains
one-parametric subgroups, each ranging over parallel velocities.

But the six-parametric transformations Z(v, ft) do constitute a
group, i.e.

for any pair of velocities and any pair of versors, and hence, in
particular, also for ftj=i, ft2=i, as in our case. For non-parallel
velocities, then, our Z(v2)Z(v1) is not again an Z(v), but it is an

RESULTANT TRANSFORMATION 169

Z(v, 12) with a certain space-rotation,* to be determined. In fact,
the formulae (6) are of the form

r" = 12[er - vy/] = for - y/flv

r = y[V__L(vr)],

where 12 =f i.

A comparison with (6) will give us the four equations

(v1 (a)

(c)

From (<£), (</) we have at once the resultant velocity, of S" relative
to S,

TI [if 73 fan)]

identical with (i), which was obtained by differentiation. The
verification that 7, as given by (b\ is equal to (i-^2/^2)~^> 1S teft
to the reader. Again, the right side of (c) is what v becomes by
permutation of lt 2> so that

12v = ft[v1 + v2] = v2 + v1, (7)

and this agrees with the nature of the operator 12. For, as was
shown explicitly, the tensors of the two resultant velocities are equal ;
cf. (3). Thus, 12 turns vx =#= V2 into v2=H=v1. The equation (7),

* In four-dimensional language the case under consideration may be expressed
as follows. Call t the time-axis in Minkowski's world. Then Zfo) will be a
rotation in the plane t, Vj ; similarly, Z(v2) will be a rotation in the plane t, V2.
Now, if VgllVj, the resultant transformation L(v^L(^) will again be a rotation
in the plane t, Vj. But if va and V2 are not parallel, the resultant four-dimensional
rotation will also have a component ' round t,' i.e. Z(v2)Z(v1) will involve also a
pure (three-dimensional) space-rotation.

170 THE THEORY OF RELATIVITY

of course, does not by itself suffice for a complete determination of
the operator, for it states the result of its application to a special
vector v only. But we have still (a), which is valid for any vector r
as operand, i.e.

12er = €,eir + y^ Vjjfor). (a)

As to €, the reader may verify that none of the above four equations
is contradicted by assuming it to be a longitudinal stretcher corre-
sponding to v, i.e. by writing, for any r,

Then 12 will be determined by (a). In fact, take for r a vector n,
normal to the plane V15 v2, and consequently normal also to v
(which is always coplanar with v15 v2). Then (vxn) and (vn) will
vanish, and en = n, so that (a) will become

12n = €.2 tjii,

and since e15 e2 are longitudinal stretchers and n is normal to the
axes of both,

12n = n. (8)

Thus, the axis of rotation, or simply the axis of 12, is normal to the
plane vls V2, while the angle of rotation is given by (7). The
outstanding determination of the sense of rotation is left to the
reader. To resume :

The general or six-parametric Lorentz transformations Z(v, 12)
constitute a group, but the special or three-parametric transforma-
tions Z(v, i) or Z(v) do not constitute a group, though they contain
the subgroups for parallel velocities. The successive application
of two special Lorentz transformations with non-parallel velocities
vi> V2 &ves always an Z(v, 12), that is to say, it is equivalent to a
special Lorentz transformation followed by a pure space-rotation
round an axis normal to vx and V2, which turns v = vx =jj= v2 into
v2=H=v1, — the former of these vectors being given by (i), and
the latter by (2).

The above properties might be elegantly expressed in quaternionic
language, by taking instead of our Q [ ] Q the more general
operator a [ ] <£, consisting of a pair of unit quaternions a, b, whose

PARALLEL VELOCITIES 171

axes are not parallel. But this subject need not further detain
us here.

We have touched the six-parametric Lorentz group only to
elucidate the question of successive transformations, as intimately
connected with the composition of velocities. But henceforth we
shall hardly need it any more. In fact, our previous transformation
Z(v), without any rotation of the space-framework, will be found
sufficient for all physical purposes.

Having got through this, let us return to the ' Addition Theorem '
of velocities, (i), with the purpose of illustrating its meaning by a
few remarks and some simple examples.

In the first place, if both Vj and v.2 are small as compared with the
velocity of light, then, if magnitudes of second order are neglected,
(i) reduces at once to

which is the Newtonian or classical formula for the composition of
velocities.

Next, consider the simplest case of parallel velocities. Then
€^2 = 7^0, and, as in Chap. V.,

or, counting the resultant velocity positively along v1?

v- ± v»

according as V2 is concurrent with or against Vj. It will be enough
to consider the former case, for which

V-, + V^

v = '^

I +V-,

Let both v} and r., be smaller than r, say, v^ — c-m, v2 = c - n,
where m, n are positive and smaller than c. Then

,
2C - m -n + mnjc

i.e. the resultant of any velocities smaller than the velocity of light in
radio is again smaller than the velocity of light. In other words,
c plays the part of an infinite velocity, inasmuch as it cannot be

172 THE THEORY OF RELATIVITY

obtained by the accumulation of any number of velocities smaller
than c. This property, proved here for concurrent velocities, will
be expected to hold a fortiori for velocities of any direction. The
rigorous proof, to be based upon the general formula (4), is left to
the reader as a useful and interesting exercise.

Again, if one of the compounded velocities, say vlt is equal c,
then, by (9),

z> = — — - = c.

I + Vjt

i.e. the resitltant of c and of any other parallel velocity (no matter
whether it is smaller or equal to or even greater than c) is again
the velocity of light c* This result becomes obvious, when it is
remembered that in the present case the system S' becomes a
flatland, perpendicular to the direction of motion, and that V2 or
the former p' is the velocity of our particle relative to S' . The
whole path of the particle appears to the ^"-observers as a single
point of that flatland, so that, for these observers, the particle
might as well be fixed in S'.

The following is one of the most beautiful applications of Relativity
that were made in the early times of the doctrine.

To emphasize better the meaning of the various velocities, write
again, for the moment, /, v, p' instead of v, vlt v.2, so that

Now, this can be put into the form

where K, expressing the fraction of v, which is added to /', is given
rigorously by

(10)

and approximately, for moderate values of p'jc and small values
of vie, by

(»)

*The discussion of cases of non-parallel velocities, to be based upon (4), is
recommended to the reader.

DRAGGING COEFFICIENT 173

Here p' is the velocity, as observed in S', of what we have hitherto
called a 'material particle.' But in doing so, we have assumed
only that it is something that can be recognized and watched in its
changing position. Its being ' material ' or not, mattered, in fact,
but little. We might as well have spoken from the beginning of
any comparatively permanent complex of sense-data, distinctly
localizable in the S- and .S'-spaces. Thus, if p' be the velocity of
propagation or transfer of anything that can be watched,* from the
^'-standpoint, and if v be the velocity of S' relative to 5, then /,
as given by (90), will be the corresponding velocity of propagation
or transfer, from the 5-point of view, and the above K will be the
dragging coefficient of S' (if it be empty except for the framework),
or, as the case may be, of the bodies or media carried along with S '.
If, for example, S' is attached to a column of air blowing uniformly
past an observer resting on earth (S), and if /' be the velocity of
sound relative to S' (and consequently, by the principle of relativity,
also the velocity of sound as would be obtained by our ^-observer
in quiet air), then (n) will be the dragging coefficient of air for
sound. In this case p'\c is of the order 3-3 . io4/3 • io10=Fio~6,
so that K differs from unity by little more than one millionth, and
we have a sensibly (though not rigorously) full drag of sound by
air. Similarly, for light! propagated along a column of flowing
water, as in Fizeau's experiment, if/' be its velocity relative to the
water and taken from the ^'-standpoint (and hence also the velocity
of light in stationary water from the standpoint of an ordinary or
S-observer), formula (n) will express the drag of light by water.

* ' Propagation,' as here defined, does not necessarily involve any material
medium as the ' substratum ' of the thing to be recognized and watched in its
migrations, the only requirement being the possibility of its being watched so.
Thus, we may have * propagation ' of a distortion along a rope, or of sound waves
in air, or of electromagnetic ' disturbances ' through empty space as well as
through glass or water. The process of detecting and watching the waves or
disturbances may be immediate in some, and very indirect in other cases, but this
does not bring in any essential differences.

t In this case we can imagine an irregular train of light waves or a solitary
wave or a sufficiently thin electromagnetic sheet which can be watched, at lease
theoretically. And if we wish we can reduce this case to that of the motion
of a * material particle,' by placing such a particle (in our imagination, of course)
in that sheet and by requiring it to be permanently illuminated ; for then it will
have to move just as quickly as light in the medium in question. This is Laue's
device, slightly modified. But I do not think that such a reduction to the motion
of something tangible is seriously needed.

174 THE THEORY OF RELATIVITY

The only difference is that in this case the value of p'jc is no longer
exceedingly small as for sound and air, and this is why the case is
of considerable physical importance. For water in ordinary con-
ditions p'jc is as great as 3/4, and it approaches unity even more
nearly for optically 'rarer' media. Generally, if n be the corre-
sponding index of refraction, we have p'\c= i/;/, so that (n) gives
at once

and this is the famous dragging coefficient of Fresnel, which occupied
so much of our attention in the early part of this volume, and
which was found to be in such good agreement with experiment.

Thus, Fresnel's formula, which on the ground of the electron
theory appeared as the outcome of a rather complicated play of
minute particles, follows here as a simple consequence of the
fundamental theorem of relativistic kinematics, quite independently
of any theory of the structure of matter.

Notice that the above is but an approximate value of the dragging
coefficient, and that its rigorous value would be, by (10),

where p = vjc. But for the present Fresnel's formula, considering
the technical difficulties of the measurements, is more than sufficiently
accurate. Remember that in Fizeau's experiment, as repeated in an
improved form by Michelson and Morley (p. 41), the water was
flowing with a velocity of 8 metres per second, so that /3 was of
the order io~8, while the observed value of the drag could be
trusted to hardly more than two decimal figures. I do not know
what possibilities lie in canal rays. At any rate the experimental
discrimination between (12) and the Fresnel formula is a problem
reserved for the future.

As a further example of composition of velocities, let us consider
the case of perpendicular components. Returning once more to the
notation adopted in the general formula (i), we have in the present
case (v1v2) = o and €1v2 = v2, so that the resultant v = v12 of vt
followed by V0 becomes

v
7i

PERPENDICULAR COMPONENTS
Similarly, the resultant of V2 followed by Vj will be

175

(14)

In Fig. 14, in which OANB is a rectangle, the former of these
vectors is given, in absolute value and direction, by OC, and the
latter by OD, while the diagonal ON represents the Newtonian
resultant. As was already remarked, the absolute values of the

B

relativistic resultants V12, V21 are equal to one another, the square
of each being in the present case given by

instead of which we may conveniently write

p-ft'+ft'-AW,

or also, as a particular case of (b\ p. 169,

(16)

To obtain the angle f = COD enclosed by the two resultants, take
their scalar product and divide it by v~. The result will be

cos c =

(17)

Thus, for vl = z>2 equal \> £, T9^ of the velocity of light, the angle C
would be, in round figures, i°, 8°, 43° respectively, or more accurately
i° 10', 8° 13', 42° 54'.

To use Sommerfeld's illustration,* if we have a rectangular ruler,
whose edges coincide initially with OA and OB (Fig. 14), and,

*A. Sommerfeld, Verhandlnngen der Deutschen Phys. Ges., XI. 1909, p. 577.

176 THE THEORY OF RELATIVITY

while it is moved relatively to the paper (S) horizontally with the
velocity vlt the point of a pencil is led along the vertical edge with
the velocity z>2 relative to the ruler (S')} then the pencil will draw
the line OC, e.g. the segment OC in unit time (S-time). On the
other hand, if the ruler is moved vertically with velocity V2 and the
pencil is led along its horizontal edge with velocity vlt the point of
the pencil will draw the line OD. According to classical kinematics,
the line drawn would be in both cases the diagonal of the rectangle.
Notice that from the paper-standpoint the velocities to be com-
pounded are : in the first case OA and A C (not AN), and in the
second case OB and BD (not BN\ In the old kinematics there
was no question of discriminating between the paper- and the
ruler-standpoints.

So much to explain the true meaning of Vj =|j= V2 , as distinguished
from v2 =|f= Vj .

The space in the ordinary sense of the word, or the space of
positions being assumed Euclidean in both the old and the new
theory, the space representative of velocities, or what is shortly
called the kinematic space, is again the Euclidean space in classical
kinematics, but non-Euclidean in relativistic kinematics. In order
to represent the resultant V12 on the same Euclidean plane drawing
with the component velocities, we had to cut off from v2 the piece
CJVt and similarly, in constructing V21 we had to cut off from va
the piece DN. If we want to obtain the resultant by a triangle
construction without cutting off anything from the segments repre-
senting the component velocities or any functions of each of these
velocities alone, then we have to use a non-Euclidean space, e.g.
Lobatchewsky's and Bolyai's space of constant negative curvature,
or, as it is appropriately called, a hyperbolic space.*

In short, the relativistic kinematic space is a hyperbolic space.

*This was first pointed out explicitly by V. Varicak, Phys. Zeitschrift, Vol. XI.
1910, pp. 93, 287, 586; cf. also Jahresbericht der deutschen Math. Vereinigung,
Vol. XXI. 1912, p. 103, where all his contributions to the subject are collected.
But it must be noticed that materially the discovery was made previously, in 1 909, by
Sommerfeld ( Verh. deiitsch. Phys. Ges., XL p. 577), when he proved that the
relativistic formulae for the composition of velocities are ' no longer the formulae
of plane but those of spherical trigonometry (with imaginary sides),' e.g. those which
are obtained from the usual ones by replacing the real radius A* of the sphere by i/v*,
— and the identity of these formulae with those valid for triangles in Lobatchewskyan
space has been well known for a long time. In fact, this identity was pointed out
by Lobatchewsky himself.

COMPOSITION OF VELOCITIES 177

To see this, take, for simplicity, the above case of Vj-LVg.
Denote the angle contained between Vj and the resultant v = v12,
i.e. the angle AOC of Fig. 14, by #2. Then, by (13),

and, by (16), 7 = 7i7-2-

Now, instead of the absolute value of each of the velocities,
introduce the corresponding imaginary angle o>,

w = arctan (i/?),

as denned by (10), Chap. V. Then y = coso>, fiy = -tsinw, and
the above pair of formulae will become

/, tan w.,
tan V9~— - -

and these are the known formulae of spherical trigonometry for a
right-angled triangle, whose sides and hypothenuse are o^, o>2, <o
and whose angle opposite to o>2 is 02, the only difference being that
here all the sides are imaginary. This is the property remarked by
Sommerfeld (cf. last footnote).

Next, to get rid of the imaginary sides, introduce, for each velocity,
instead of w the real angle a, as defined by (20), Chap. V., such that

ta.nha = /3 = v/<:. (18)

Then, as was previously noticed, w = ia, and, since
sin (to) = i sinh a, cos (ta) = cosh a,
the above formulae become at once

cosh a = cosh at . cosh a.2

Now, these are exactly the formulae for a right-angled triangle in
Lobatchewskyan or hyperbolic space.* Thus, if al and a.2 (Fig. 15)

*Cf. N. I. Lobatchewsky's Zwei geometrische A bhandlungen, translated from
Russian into German and edited by F. Engel, Leipzig, 1898. Also 'Non-
Euclidean Geometry,' by Frederick S. Woods, in Monographs on Topics of Modern
Mat hematic s, etc.) London, 1911, or R. Bonola's Non-Euclidean Geometry \ trans-
lated by H. S. Carslaw, Chicago, 1912.
S.R. M

178 THE THEORY OF RELATIVITY

are segments of geodesies or shortest lines in hyperbolic space,
representing the component velocities, the shortest line a, completing
the triangle, will represent the resultant velocity, as regards both size
and inclination, 0.2. The same property may be proved to hold in
general, i.e. for component velocities including with one another
any angle. Here it will be enough to give the length of a.

90°

FIG. 15 FIG. 16.

Denoting by ?r - 0 the angle vl , V0 , so that 6 itself is the angle
opposite to a (Fig. 16), we have

so that our previous formula

becomes at once

cosh a — cosh a^ . cosh a.2 - sinh a} . sinh a2 . cos 0. (20)

The determination of the angle #9, by means of the general formula
(i), is left to the reader.

Notice that, as long as we are concerned only with two velocities
and their resultant, we have no need of three-dimensional hyperbolic
space. What we want then is a Lobatchewskyan plane or a surface
of constant negative curvature. Now this may be easily procured
of any size in Euclidean space. Models of such a surface, known
as a pseudosphere, which is a surface of revolution,* belong now to
the outfit of many mathematical class-rooms. Our last two figures
must be imagined to be drawn on a pseudosphere (which certainly
has nothing more imaginary about it than the page on which Figs.
15 and 1 6 are drawn), the curved sides of our drawings being as
straight as possible on such a surface. Thus, having at our
disposition a pseudosphere, we could study at our leisure the non-

* See, for instance, Bonola's book, just quoted, p. 132.

COMPOSITION OF VELOCITIES 179

commutativity and all the remaining properties of the addition
of velocities. In this way the relativistic rules of the composition
of velocities could be made accessible even to all those who do not
like to think of hyperbolic, and other non-Euclidean, spaces.

It has been proposed by Dr. Robb* to call our above a, as
defined by (18), that is

« = arc tanh- , (21)

the rapidity, corresponding to the velocity v. It seems a very con-
venient name for the purpose. Using it, we may briefly restate the
above result as follows :

Any two rapidities are compounded by the triangle-rule in
hyperbolic space.

Whence also : the resultant of any number of rapidities arranged
in a chain in hyperbolic space, is the geodesic or the straight line
of that space, drawn from the beginning to the end of the chain.

Notice that if rapidity is to involve 'direction' as well as size
or absolute value, it has to be considered as a vector localized in its
own line^ i.e. in a Lobatchewskyan straight line or shortest line
upon our pseudosphere. In connexion with this we have only
the triangle-rule, and not the parallelogram-rule, as in Newtonian
kinematics. There are no parallelograms in hyperbolic space or
upon a pseudosphere, any more than upon a sphere. To express
that direction is involved, we may write for the rapidities a15.a2, etc.,
and use the ordinary sign + for their addition, keeping in mind
that each of these rapidity- vectors can be shifted only along its own
line, and, consequently, that their addition is non-commutative,
unless a1} a., are on the same line. Thus, the rapidity aj + a2
(Fig. 17) is AB, while a2 + ax is CZ>, which, though of the same
length, is on a different line.

* Alfred A. Robb, Optical Geometry of Motion, Cambridge, W. Hefier & Sons,
1911.

i8o

THE THEORY OF RELATIVITY

Remembering that tanh« = (ea - e~a)/(ea + e~a\ we can write, instead

Of (2 1),

For small values of ft we have, up to quantities of the second order,
a = ft = v/c, so that for small velocities the corresponding rapidities
are small fractions, of the order of ft and the Lobatchewskyan
triangle becomes a Euclidean triangle, as in classical kinematics.
It seems worth mentioning that to unit rapidity corresponds a huge
velocity, amounting to J of the velocity of light ; more accurately,
we have

^ = •7616 for a= i.

90

From (210} we see most immediately that to the velocity of light
itself corresponds an infinite rapidity,

a = oo for ft = i .

Now, if two sides of a pseudospherical triangle are finite, its third
side is also finite. Thus, our previous statement, that the resultant
of any velocities smaller than that of light is again smaller than the
velocity of light, is reduced to an obvious property of hyperbolic
triangles.

To close the discussion of this beautiful subject, but one remark
more. Lobatchewsky's II (a), the angle of parallelism for the length a,
as explained by Fig. 18, is related to the above hyperbolic functions,
for any a, as follows :

— -, —
cosh a

, tan II (a] = ^— : — (22)
^ ' smh a

COMPOSITION OF VELOCITIES 181

Thus, equations (19) can be written, in terms of ordinary trigono-
metric functions of the respective angles of parallelism,

(»s>
'2), J

tan 02 = tan II (aj . cos II («2)

which is the original form of Lobatchewsky's own formulae, for a
right-angled triangle. Similarly, the general formula (20), will become

. TT/ x sn «, . s2 / v

smll(0) = - / y - / f - 7,1 (24)

i - cos n(0j) . cos II(tf2) . cos 0

which is Lobatchewsky's fundamental formula. The unit of length
here adopted is that employed by Lobatchewsky, i.e. that length
whose negative reciprocal square is the curvature of the representative
hyperbolic space, or the curvature of the pseudosphere upon which
the triangles are to be drawn. Thus, if we take for that purpose
a pseudosphere of curvature - i/ioo cm2., a segment of its geodesic
10 centimetres long will correspond to the rapidity a=i, and con-
sequently will represent the velocity -76 c which is a little above
the velocity of light in water.

Instead of (18), we shall now have, by the second of (22), and
omitting the unnecessary argument,

For very small values of /5 the angle of parallelism II is nearly a
right angle, as in a Euclidean plane. Thus, for the earth's orbital
motion /?=io~4 and 11 = 89° 59' 39 '4» so tnat tne departure from
Euclid amounts only to 2o"-6. But if we turn to swift electrons,
as observed in kathode rays and /2-rays of radioactive substances,
the angle of parallelism is very considerably reduced. For ft = -go
and -95 (Kaufmann observed even -99 and more) I find 11 = 25° 5°
and 1 8° 12' respectively. At the limit, for light -velocity, the angle of
parallelism would vanish altogether.

CHAPTER VII.

PHYSICAL QUATERNIONS. DYNAMICS OF A PARTICLE.

THE importance of the study of world-vectors or of physical
quaternions for relativistic investigations is obvious. For, if the
form of the laws of physical phenomena is to be preserved by the
Lorentz transformation, they can involve besides the time and
the coordinates, and, of course, besides any invariants, only such
sets of magnitudes which, caeteris paribus, bear in any of the
legitimate systems the same relations to its time and coordinates as
in any other of such systems. Therefore, physical quaternions (or
whatever mathematical form we may choose for tetrads of magni-
tudes transformed like /, x, y, z and of sets of magnitudes derived
from them) constitute, as it were, the building material of the
modern relativist. And what is most important to keep in mind, is
that he cannot use any other material. For if he did, he would be sure
to infringe against the fundamental principle of the whole theory.

To try to describe in a few abstract sentences the way how this
material is procured and how it is used, would be a vain attempt.
The reader will see it best from particular cases.

As yet we had, properly speaking, only one physical quaternion,
which we made the standard of such quaternions, e.g. the position-
quaternion

(i)

This was transformed into q by the operator Q [ ] Q. If any
quaternion X was transformed into X' by the same operator, we
wrote X~q, and if it had also, like -7, an imaginary scalar and a
real vector, we wrote X~q, and called X a physical quaternion.
Such was our definition given in Chap. V., entirely equivalent to
that of a four-vector.

VELOCITY-QUATERNION 183

Now let us look for other physical quaternions. An indefinite
number of such can be obtained at once from q itself.

In fact, let q belong, say, to a material particle at a given instant /
of its history. Let the particle move about in an arbitrary manner,
and let p be its instantaneous velocity in S. Then its position-
quaternion at the instant t+dt will be q + dq, and this as well as q
will certainly be a physical quaternion. And since Q [ ] Q is
distributive (or since the Lorentz transformation is linear and homo-
geneous), the difference of these two quaternions, i.e.

dq = dl+d* = [ic+ p] dt,

will again be a physical quaternion, ~^. Therefore, as we know
from Chap. V., its tensor

will be an invariant. Divide it by ic; then

will again be an invariant. Its value will be real, provided that
p is not greater than c. And since dq is a physical quaternion, we
shall have also

r-J«ft (3)

that is, Y will again be a physical quaternion. Let us call it 'the
velocity-quaternion of the particle in question. Its developed form is

where p is the ordinary vector-velocity of the particle, justifying
the above name.

The plain meaning of our result is that Y = Q YQ, i.e. that

icyp and pyp
are transformed as / and r, or, what is the same thing, that

yp and pyp
are transformed like

/ and r.

1.84 THE THEORY OF RELATIVITY

Using this, the reader will obtain at once the addition theorem of
velocities, identical with (i«), Chap. VI., along with the formula

(identical with (b\ p. 169), which is a consequence of that theorem.
Thus, the relativistic rule for the composition of velocities is implied
in the statement that Y is a physical quaternion.

The infinitesimal scalar dr, as defined by (2), deserves special
attention. For / = o it reduces to dt, the element of ordinary
5-time, but is, in general, smaller than dt. It has the advantage
of being an invariant, which dt is not. In other words, the value
of dr is independent of the choice of our standpoint, being equal
for all legitimate systems. It belongs to the particle. The same
property will obviously hold for

-H

dt

where the integral is taken along any portion of the particle's history,
or along any segment of its world-line, from an arbitrarily fixed
initial point to the variable end-point. The parameter r, thus defined,
may be called, after Minkowski, the proper time* of the particle.
The velocity p of the particle, entering into each element of r by
its square, may, in general, vary from instant to instant, as regards
both absolute value and direction. If the particle is fixed in S, its
proper time is the ordinary time / of the system £. And if the
particle moves uniformly in S, we can imagine a system Sf in which
it will be at rest. And then the proper time of the particle will
become the ordinary time of that system.

The velocity-quaternion may now be described as the derivative
of the position-quaternion with respect to the proper time of the
particle. It will often be convenient to use the dot for this
differentiation. Thus, Y— q.

The name corresponding to Y in the language of four-dimensional
algebra would \*e four-velocity, \ and its matrix-form would be simply,

by (30),

19\Pm A» /«» «?!•

* Eigenzeit.

t Minkowski's Bewegungsvektor, Laue's Vierergeschwindigkeit.

VELOCITY AND ACCELERATION 185

Remember that dr, as originally defined, was simply the tensor
of dq divided by ic. The tensor of the velocity-quaternion is,
therefore,

TY=iC. (4)

We know, from Chap. V., that the tensor of every physical quaternion
is an invariant. In the present case this knowledge does not
furnish us anything new. For c is, by the fundamental assumptions
of the theory, a universal constant. The norm of Y being negative,
namely equal to -c~, the velocity-quaternion is always time-like.
In Minkowski's language we should say that the four-velocity is
along the world-line of the particle in question.

Since Y is a physical quaternion and r is an invariant,

will again be a physical quaternion which, for obvious reasons, may
be called the acceleration-quaternion. So also will dzqjdr^^ etc. be
physical quaternions, each ~q, and obviously also d^di*, etc.,
each ~^c- But of all these derivatives of q we shall hardly need
more than the first two, containing the velocity and the acceleration.
Let Yc be the conjugate of K Then, by Quat. 7, we can write
for its norm the product YYe or also SYYC, and consequently,
instead of (4),

Differentiating this with respect to r, we have

ZFc+yZe = o, (6a)

or also

5ZKc = o, (6)

which says precisely the same thing as (6a).* Such then is the
relation which holds always between the acceleration- and the
velocity-quaternion of a particle. Using the developed form ^ =
we should have, correspondingly,

and (rr) + /7=o,

* In fact, the reader will find at once that, for any pair of quaternions a, b,
abc + bac — 2Sabc —

1 86 THE THEORY OF RELATIVITY

or, in a still more developed form,

i2+j,2 +

and

In four-dimensional language, as explained in Chap. V., the last
formula would read : The four-acceleration is always normal to the
four-velocity and, consequently, to the world-line of the particle, — a
famous statement of Minkowski. This cardinal property finds then
its short quaternionic expression in (6). Observe that the left side
of that equation is the same thing as Sommerfeld's scalar product
of the corresponding four-vectors. But the invariance of such
expressions is seen more immediately on the quaternionic scheme.
In fact, remembering that QQe= QCQ= TJ we have, by Quat. 6,

Next, as regards the transformational properties of the acceleration.
These are entirely determined by saying that Z=ict + Y is a physical
quaternion. For this means simply that /, r are transformed like
/, r. If, therefore, S' be a system moving relatively to S with the
uniform velocity v, we have, according to (i'^), Chap. V.,

where the subscripts are to remind us that y, e are to be taken for
the velocity v. The dots denote, on both sides, differentiation with
respect to the same variable r. For, as the reader already knows,
d-c' = d-r. There is no difficulty in developing these formulae and
thus finding the ordinary acceleration

in terms of a' = cty ' jdt' and p' (or vice versa), for any pair of legitimate
systems S, S' picked out at random. But this would hardly be
worth the trouble.

To see the plain kinematical meaning of the second derivatives
with respect to r and hence of the whole acceleration-quaternion,
we have to place ourselves at a standpoint which bears the simplest

REST-ACCELERATION 187

possible relation to the moving particle itself. Let us then take
for 5' that particular system of reference with respect to which
the particle is instantaneously at rest. In other words, let S' be a
system whose uniform velocity v, relative to S, is equal in size and
direction to the instantaneous velocity of the particle, i.e. to the
value of p at a given instant of its history. Then, at that instant,
p' = o and Y = y (/')=i. Now, we had, generally,
Therefore,

y, ,dy' dy' i ,».,dp'

1 = = = = 0

or /' = o, as might have been expected, and in a similar way,

so that Z' = /' + r', the acceleration-quaternion relative to S', for the
instant in question, is simply

Z' = a', (8)

i.e. equal to the ordinary acceleration of the particle with respect
to S'. Since S' is that particular system of reference in which the
particle is instantaneously at rest, it may be called the rest-system
and the corresponding a' the rest-acceleration of the particle.*

Thus, the scalar part of the acceleration-quaternion Z' vanishes
identically, and its vector part is equal to the rest-acceleration, f
Consequently, TZ' = a. And since the tensor of every physical
quaternion is an invariant, we have also, for any legitimate system S,

TZ=a. (9)

In words, the tensor of the acceleration-quaternion is equal to the
absolute value of the rest-acceleration of the particle. It acquires
thus an immediate kinematical meaning. At the same time formulae
(7), in which we have now to write v = p, give us, for the system S
which in a certain sense is an unnatural system of reference,

r = €a' (10)

* In German, A'z^beschleunigung.

f This result could be foreseen. In fact, the ' ordinary ' time of our particular
S' coincides, in its element in question, with the proper time of the particle.

1 88 THE THEORY OF RELATIVITY

and /=r~2y(pa'), so that the whole acceleration-quaternion may be
written :

„ dY i

/ x

Here, p is the velocity of the particle relative to S ; y = yp, and
the stretcher e = €p, of ratio yp, acts along the instantaneous direction
of p or tangentially to the path of the particle. Thus, in Cartesians,
if the tangent to the path of the particle, drawn in the direction of
its motion, be our instantaneous :r-axis,

and ct=/3yax'. If the j'-axis be taken in the osculating plane of
the path, then H = o. Since we already know, by (9), that

the formula for / becomes superfluous.

Finally, to express r = d2i/dT2 in terms of the ordinary ^-accelera-
tion a=d?p/*#, remember once more that dt/dr^y. Since, by the
definition of y,

dy i „ dp

the result will be

r = 7 ^ = 72 [a + ~ y2p(pa)] = y2 [a + /5Vu(ua)],

where u is the unit of p. Now, i + /22y2 = y2, identically. Therefore,
the bracketed expression is the vector sum of the longitudinal part
of a magnified y2 times and of its unaltered transversal part, or
simply the result of a double application of the stretcher e. Thus,
ultimately,

r^-yVa, (12)

whence also, by (10),

y2ea = a', (13)

giving the connexion between a and the rest-acceleration. Or, in
Cartesians, with the above choice of axes, for the longitudinal and
the transversal components of r,

x = y*<*x, y = fay, z = yzaa, (12 a)

and

HYPERBOLIC MOTION 189

By (13) we have also, writing pjc = /?,

/^sm(i>) = a', (14)

which is merely a developed form of (9). In fact, the Hgftlhand
side of (14) is seen, by (n), to be identical with TZ

The simplest case of motion of a particle occurs when a is
permanently nil, and consequently also a = o. This is, as in classical
kinematics, the trivial case of uniform rectilinear motion. Such motion
preserves its character in all legitimate systems. In fact, owing to the
linearity of the Lorentz transformation, any motion which is uniform and
rectilinear with respect to one of these systems will be so relatively
to any other of them. A straight world-line will remain straight.
The next simplest kind of motion, which also preserves its character
in all such systems of reference, occurs when the non-vanishing rest-
acceleration is constant in size and direction^ i.e. when da,'ldr' = o, and
hence also </a'/dT/=o. Then, by (13), the vector y'2ea is constant
in S, that is to say, independent of /. But since the axis of the
stretcher e, or the #-axis in (130), instead of being fixed, is at every
instant tangential to the path of the particle, which may be curvilinear,
it does not follow that even the direction of the acceleration a will
be constant in S. Thus, the general case of such motion, which is
the counterpart of the uniformly accelerated or parabolic motion of
classical kinematics, would still be fairly complicated. The simplest
sub-case, which also will show best the characteristic properties of
this kind of motion, occurs, of course, when the particle moves on
a straight line. Let this be our jc-axis. Then, by (13),

ft 2(72-0*
whence, counting the time / from the instant at which / = o,

Or we may write, equivalently,

190

THE THEORY OF RELATIVITY

Thus, as long as at is small in comparison with the velocity of
light (whether before or after the instant when the particle was at
rest in S), we have, approximately, p = a't, and x = \a'fi + const.,
as in the Galilean free fall. But after a sufficiently long time the
neglected terms begin to make themselves sensible, and the velocity
of the particle, instead of increasing beyond all limits, tends
asymptotically to the velocity of light. In fact, we have from
(15), for any given a, p — +c for /= + oo.

Integrating once more, and choosing the origin of x so that,
for /=o, x = x0 = c2/a, we obtain

-2\2

(16)

Thus, the world-line of our particle, in rectilinear motion with
constant rest-acceleration, is an equilateral hyperbola (Fig. 19), the

length of whose semiaxes is equal —,. This motion has therefore

a

been called by Born, who was the first to study it, hyperbolic motion.*
The asymptotes AO, OA correspond, as in a previous figure, to the
velocity c, directed towards and away from the origin. The particle
arrives from x = oo with light-velocity, moves up the #-axis with
ever-diminishing velocity towards P, the vertex of the representative
hyperbola, where its velocity is nil. Then it turns and moves down
the #-axis with increasing velocity, which again tends asymptotically
to the light-velocity. The larger the value of the rest-acceleration a\
the more does the hyperbola penetrate into the angle AOA, and the

* In German, Hyperbelbewegung. M. Born, ' Die Theorie des starren Elektrons
in der Kinematik des Relativitatsprinzips,' Ann. d. Physik, Vol. XXX. 1909, p. I.

HYPERBOLIC MOTION 191

more sudden is the passage of the particle's velocity from - c through
zero to +c. Taking, instead of S, another system of reference 5"
which moves uniformly along the jc-axis, and whose origin coincides
with O at the instant /" = /=o, we shall have again equation (16)
for the new variables. For, x^-c^t'2 is an invariant, and so is the
acceleration a, by its very definition. It is this we meant by saying
that the considered kind of motion preserves its character in different
systems of reference, — a property which is not shared by the
Galilean uniformly accelerated motion to which would correspond
a parabola as world-line. Remember that in classical kinematics
there was no question of discriminating between the ordinary
^-acceleration a and the rest-acceleration a.

We may mention here that the hyperbolic motion is particularly
interesting in connexion with the theory of the relativistic 'rigid7
body. But its chief importance lies herein that any variable
motion can be closer approximated by it than by uniform motion.
In other words, any curved world-line can be brought into closer
contact with a hyperbola than with a straight line. There is for
every point P of such a world-line a hyperbola of closest contact
with the world-line, which plays the part of the familiar circle
of curvature and which was called by Minkowski the hyperbola
of curvature. If O be the centre of this hyperbola (whose vertex
is at P), then the four-acceleration will be given by the world-
vector drawn in the direction OP and having the absolute

£

value c-jOP, or = — -• In fact, as we have just seen, the last

expression is simply equal to a, and this again was seen to be
identical with TZ, or with the size of the four-acceleration which
was always normal to the world-line. Remembering, on the other
hand, that T Y= ic, or that the square of the four-velocity is equal
to - c-j the reader will at once perceive the perfect analogy between
the above property of c^jOP and the familiar formula: normal
acceleration = square of velocity divided by radius of curvature. It
will also be noticed that, the square of the four-velocity being
negative, the four-acceleration is directed away from the centre O
of the osculating world-hyperbola, while in that more familiar case
it is towards the centre of (the circle of) curvature of the path in
three-dimensional space. But enough has now been said about the
hyperbolic motion, in illustration of the use of the relativistic tetrads,
Y and Z.

192 THE THEORY OF RELATIVITY

Such then are the properties of the velocity- and the acceleration-
quaternion. These being simply the (first and second) derivatives
of the position-quaternion q of a particle with respect to its proper
time, our above considerations had a purely kinematical character.
Although we have spoken of q as defining the position and the date of
a 'particle,' the latter could mean anything which can be recognized
at all and watched in its varying position. Of course, if this is to
be possible, the ' particle ' must have some or other characteristics of
its own. But these must not necessarily be quantitatively measurable,
to say nothing of their being constant in time or equal for different
standpoints or systems of reference. The moving thing in question
might have no such characteristic at all.

But let us suppose there is a certain magnitude of such a kind,
that there is, more especially, a scalar coefficient belonging or
attached to the particle and fulfilling the latter condition, i.e.
invariant with respect to the Lorentz transformation. Denote it by
m, without yet giving it any name. Then mq, mY=mq, mZ=mY,
and so on, will all be physical quaternions, and, consequently, each
of them may be employed, along with other physical quaternions,
for relativistic purposes, i.e. to write down laws of motion of the
particle. Such laws would be admissible, in that sense of the
word, that they would not infringe against the principle of relativity.
But this does not imply, of course, that they will be obeyed by
Nature. If such laws or equations are to be of any use for the
physicist, and if they do not happen to cover an entirely unexplored
ground, they have to coincide, roughly at least, and for ordinary
circumstances, with what is otherwise known to hold in experience.
In the present case we shall require that the relativistic equations
of motion should coincide, approximately for small velocities, or
rigorously, when referred to the rest-system, with Newton's second
law of motion.

Keeping this in mind, let us see what are the consequences of
assuming, as the equation of motion of our particle,

First of all, since the left-hand member is ~ q, the right-hand
member X, which is to be considered as a given function of position,
and, in general, also of velocity and of time, must again be a
physical quaternion. This fixes the transformational properties of

DYNAMICS OF A PARTICLE 193

the quaternion X, and implies that it has an imaginary scalar and
a real vector; the coefficient m being supposed real.

By its construction, equation (17) will preserve its form in all
legitimate systems of reference.

Remembering that *f//dr = y, write, instead of (17),

dmY i

and denote the imaginary scalar of y~lX by iv and its real vector
by N, i.e. put

-X=iv + N. (18)

7

Then (17) will split into the vector and the scalar equations

d

where p = dijdt is the ordinary velocity of the particle relative to S,

and 7 = (i-/2/'T--

Written for the rest-system, which we shall again denote by 5',
the first of these equations becomes at once

Midi = m -r-y = N ,

i.e. identical with the classical equation of motion of a particle of
mass m under the action of the impressed force N'. Thus, the
above requirement is fulfilled. In view of this property, the
coefficient m is called the rest-mass of the particle.* The ordinary
force, N' in the rest-system and, generally, N in any legitimate
system S, is called the 'Newtonian force' in distinction from yN,
the vector part of X, which is the ' Minkowskian force.' For reasons

* Lorentz, Phys. Zeitschrift, Vol. XL 1910, calls m the ' Minkowskian mass,'
and VJ$T=7N the 'Minkowskian force,' since (17), with constant m, is the
equivalent of the four equations of motion given by Minkowski ; Grundgleichungen,
Appendix, formulae (22). The four-vector corresponding to the whole quaternion
X is Minkowski's ' moving force (bewegende Kraft).' Einstein used the Newtonian
force; his three equations of motion are identical with the first of (ija). Cf.
Einstein?s paper injahrbuch der Radioaktivitdt tind Elektronik, Vol. IV. 1908,
pp. 411-462, formulae (li).

S.R. N

194 THE THEORY OF RELATIVITY

which will appear when we come to consider the ponderomotive
actions of the electromagnetic field, we shall have to consider the
former and not the latter as the force acting upon the particle.
The second of (ija) becomes, for the rest-system,

dm v

As will be seen in Chapter IX., there are reasons for admitting
that even the rest-mass may vary with time. In fact, this will,
in general, be the case when the internal state of the particle varies
during its motion. But, to simplify matters, let us suppose that
the particle's internal state is kept constant. Then its rest-mass m
will be constant in time. This implies v' = o, so that the whole
quaternion X will be reduced, for the rest-system, to

X' = N\ (18')

and we shall have, for any legitimate system S,

W, (19)

where JV" is the absolute value of the (Newtonian) force as estimated
from the standpoint of the rest-system.

With this supposition of a constant rest-mass, equation (17) becomes

m— = mZ= X. (20)

ar

Now, by (6), SZFc = o, and consequently also

SXYc = o, (21)

or, in developed form, by (30) and (18),

Hence, by the second of (170), which is simply the scalar part
of (20),

Thus, (Np) being the activity of the force N, the scalar part of
the quaternionic equation (20) expresses the principle of energy

DYNAMICS OF A PARTICLE 195

(principle of Vis-viva), giving for the kinetic energy of the particle
the value

T= me2 (y + const.).

If we require that, for/ = o (i.e. for 7=1), 7^ = 0, we have to put
const. = - i. Ultimately, therefore, the kinetic energy of the particle,
moving with the velocity p relative to S, becomes

T=mt*(y-i) = mc*[(i-P)-*-il (23)

or, developed in a series,

For small velocities, this reduces sensibly to the first term
which is the classical value of the kinetic energy, since in this case
the rest-mass becomes sensibly identical with its lvalue.

The above expression of the kinetic energy was first given in
Einstein's fundamental paper of 1905. An alternative, remarkable,
form of (23), due to Minkowski, is

and reads as follows : the kinetic energy of a particle, as estimated
from the ^-standpoint, is the product of its rest-mass by the square
of the light-velocity and by the proportionate gain of the .S-time
with respect to the particle's proper time.

Let us now consider the vector part of the quaternionic equation
of motion, or the first of (i 70). This, which holds also for a variable
rest-mass, may be read in the usual way : rate of change of momentum
= force. Then the momentum of the particle, of rest-mass m, will be

(24)

Thus, to obtain the momentum we have to multiply the ordinary
velocity p of the particle by my, and not by m. Some authors call,
therefore, my the 'ordinary mass' of the particle. But we have
rather to avoid so many different names. It is quite sufficient to
know that m, the rest-mass, enters in a certain way into the expression
of momentum, and in a certain way into that of kinetic energy.
The momentum-quaternion, which is always a physical quaternion, will
simply be mY.

196 THE THEORY OF RELATIVITY

Next, to see the properties of m with respect to the ordinary
acceleration a = ^2r/^/2, return once more to the assumption of
constant m, and write the first of

di m ..
w-r- = -r = N.
dt y

Then, by (12),

= N, (25)

where, it will be remembered, e2 is a stretcher of ratio y2 acting
tangentially to the path. Thus, the force, though always contained
in the osculating plane, will, in general, differ in direction from the
acceleration. Instead of the old * mass,' which was simply a scalar
factor converting the acceleration a into the force N, we have now
the linear vector-operator

my . e2.

Or, splitting the acceleration into its tangential and normal (or
longitudinal and transversal) components, ax, av,

mys.ax = lVx, my.ay = Ny. (250)

This result is expressed by saying that the particle has the longitudinal
mass

o m t s \

mt=mf=^rwf'

and the transversal mass

Wl / x- 7\

'

For vanishing velocities both of these masses become identical with
the rest-mass of the particle. With increasing velocity the longi-
tudinal mass increases more rapidly than the transversal one. For
p = c both would become infinite. So also would the kinetic energy
of the particle increase beyond all limits when the velocity of light
is approached.

It is worth noticing here that the above ml and mt depend on the
velocity of motion in exactly the same way as the longitudinal and
the transversal electromagnetic masses of a Lorentz electron.* The

* In fact, the above formulae (260), (26b) become identical with those of Lorentz

tf2

when m is replaced by - ^— in the case of homogeneous volume-charge, and by
e* $re*K

6irciR m ^e case °^ h°m°geneous surface-charge, e being the charge and R the
est-radius of the electron. Cf. Chap. VIII.

DYNAMICS OF A PARTICLE 197

formula of Lorentz for the transversal electromagnetic mass is now
fairly well verified by experiments on electrons constituting the /3-rays.
In the early stage of such experimental research other electronic
formulae coincided equally well with the observed facts. It has
been argued therefore that the whole mass of the electron is of purely
electromagnetic origin. Now, the above relativistic formulae, giving
the required dependence on velocity, have nothing electromagnetic
about them. If, therefore, the doctrine of relativity is accepted, any
part of the observed mass of the electron may be attributed to a
non-electromagnetic origin. To obtain this we have only to give
to the electron, instead of the usual io~13 cm., a correspondingly
greater radius, reducing thus its electromagnetic mass. Remember
that what is given by observation is the total mass and the total
charge of an electron, while its dimensions remain free, in very wide
limits at least. But this subject cannot profitably be discussed here
any further.

The longitudinal and the transversal masses of a particle, defined
as the quotients of the corresponding components of force and
acceleration, may also be written, by (24), in terms of the absolute
value G of the momentum,

G dG

mt = T mi = ~dp' (27)

The first of these is simply (24) itself, and to see the truth of the
second, we have only to remember that dyjdp = yzplci* The
formulae (27) would even continue to be true if we had in the
expression of the momentum, instead of the factor my, any other
function of ft alone, as the reader may easily prove for himself.

Let us once more return to the first of equations (170), which
may be written

Multiply it on both sides vectorially by r. Remember that the
momentum coincides in direction with the velocity p = */r/*#, or
that VpG = o. Then the result will be

So that

. (29)

at

198 THE THEORY OF RELATIVITY

In words : The rate of change of the moment of momentum is equal,
in absolute value and direction, to the moment of the impressed
force, both moments being taken about O, the origin of r. This is
the relativistic equivalent of what is known in classical dynamics as
the principle of areas. The above moment of momentum is, in terms
of the rest-mass m, and r, r,

^ &Vl0»iwVr~

d-r

In particular, if the moment VrN is permanently «/7, i.e. if the
impressed force is central, we have the equivalent of the principle
of conservation of areas, that is, m being again supposed constant,

where the vector A is constant both in size and in direction, relative
to the frame-work of reference 6*. In this case the particle moves in
a plane, normal to A, as it would also according to Newtonian
mechanics. But there is the following difference. In terms of the
usual polar coordinates r, 0, we have, by the last equation,

r^-A

r 5 4

that is to say, equal areas swept out by the radius vector in equal
intervals of the proper time of the particle, and not of the ,S-time.
Using the time t of the observers fixed in S we should have

and this is variable, unless the particle happens to move uniformly
along its orbit. Such then is the relativistic modification of Kepler's
second law, valid for any central forces. For slow motion we fall
back, of course, to the ordinary conservation of areas.

Leaving, for the present, any further dynamical questions, we shall
close this chapter by developing some simple and general properties
of certain combinations of physical quaternions, independent of their
particular meaning. These will be found useful in connexion with
the subject of electromagnetism to be treated in the next chapter.
They may also have a certain interest of their own.

Let #, b, d^ etc., be any physical quaternions, each ~ ^, and, conse-
quently, #c, /£c, dc, etc., each ~$rc. What combinations of these

PHYSICAL QUATERNIONS 199

quaternions, obtained by their addition and multiplication, can be
used for relativistic purposes, that is to say, for writing down
equations which will satisfy the principle of relativity?

We need not dwell upon the sum a -f b 4- d+ . . . (or ae + bc + de +...),
which is again a physical quaternion, in the original sense of the
word, and as such, is relativistically available. But having mentioned
the sum at all, it may be good to observe that a sum of antdvariant
quaternions,* as, for example,

cannot be used. For not only is this sum not covariant with q, nor
with qn but, when subjected to the Lorentz transformation, it is
split, the two addends being torn asunder, thus

In other words, such a sum is not transferred as a whole from one
legitimate system of reference to another.

Now for the product of physical quaternions. Begin with the
simplest case of two factors. Leave aside ab which is split in the
act of transformation, thus

and pass straight on to the product of antivariant factors^ say,

H=acb. (30)

Pass from the system 5 to any other legitimate system S'. Then
H' — QcacQc - Q&Q) whence, by the associative property, and re-
membering that QeQ=i,

H'=QeHQ. (31)

Thus, the new quaternion Jf, though it is neither covariant with the
standard q nor covariant with qc, is transformed as a whole (com-
posed of constituents already admitted) and can therefore be used
for relativistic purposes. A moment's reflection will convince the
reader that such a procedure will not infringe against the principle
of relativity. And the meaning of these abstract remarks will

* Any two quaternions of the set

a, b, </,...,
or any two of the set

Oct bey dct ••• >

being covariant with one another, we may conveniently call any quaternion of
the first set anti-variant with respect to any one of the second set.

200 THE THEORY OF RELATIVITY

become plainer when we come, in the next chapter, to consider
a concrete law involving a magnitude which, in passing from S to S',
is transformed exactly as the above quaternion H. Meanwhile, let
us look for some further properties of that quaternion.

Consider Ifc, the conjugate of H. This will be, by the elementary
rule of the conjugate of a product, Quat. 8,

Now, transforming this, we get Hc = Qc&cQcQaQ> or> m exactly the
same way as above,*

HC=QCHCQ. (32)

Thus we see that

Qe[ ]Q

is the relativistic transformer of both H and its conjugate Hc, and
hence also of their sum and of their difference, i.e. also of the
scalar and of the vector parts of the quaternion H separately, say s
and L,

Now, s being a scalar, we have simply

s'=Qc*Q=*QcQ = s,

i.e. s is an invariant, as was proved before. Thus, the scalar part
of acb need not detain us any further.

What we really need for the subsequent physical application is L,
the vector part of this quaternion. This is transformed into

L'=<2CL<2, (33)

and since Q, Qc are unit quaternions, the tensor of L is an invariant,
TL' = TL, which may also be written, more conveniently,!

L'2 = IA (34)

* Here, HC is an abbreviation for (J7C)', the transformed conjugate. But
taking the conjugate of the transformed quaternion, (31), we obtain at once
(H')C=QCHCQ, so that (HC}' = (H')C, and both sides may, therefore, be written
simply HC.

t Remember that the square of the tensor, or the norm of any quaternion X
is XXC- Now, in our case, L being a scalarless quaternion, its conjugate is
LC = — L, so that its norm is simply — L2. If L were an ordinary, real vector,
we could write (instead of -L2) Z2, the square of its size or absolute value.
But since L is a complex vector, or a bivector, the above notation is preferable.
L2 is a scalar, of course, e.g. a complex scalar, as will be seen presently. We
need not put the prefix S before it, since VLL is always nil, by the elementary
definition of vector product.

PHYSICAL BIVECTORS 201

These being the transformational properties of the vector L = Vae/>,
let us see what is its structure.

Since both a and b have the structure of q, the standard of
physical quaternions, write

; .*. ac = ta — A
and b = i/3 + B,

where a, ft are real scalars and A, B ordinary, i.e. real vectors. Then

L = L1-iL2, (35)

where Lj and L2 are the real vectors

L1 = VBA, L2 = £A-aB. (36)

Thus, L is a complex vector or a bivector, — called so, since it
consists of two ordinary vectors. We had, in Chap. II., a sample
of such a magnitude in the electromagnetic bivector. The complex
invariant, (34), of L splits into its two real invariants,

L*-L* and (L^). (37)

The second of these invariants vanishes, since, by (36), Lx is
perpendicular to L2. This being the case, L = V«C^ is a special
bivector (and is equivalent to Sommerfeld's 'special six-vector').
In order to obtain the general bivector, whose two real vectors
are mutually independent, we have only to add to the above L
another, appropriate, special bivector having the same transforma-
tional properties. For this purpose we can take the special bivector
L(*>, the supplement of L, denned by L<s> = VaW>, where a^\ ^ is a
pair of physical quaternions, such that

Sa^ae = SaW&c = S#8}ac = SWe = o.

But particulars concerning the choice of a sufficiently general
supplement, as this is, need not detain us here.

Henceforth we shall denote by L the general bivector, thus
obtainable. And we shall call it, where it will be needed for the
sake of distinction, a left-handed bivector, owing to the position of
the subscript c in its transforming operator, or in the generating
quaternionic factors : ac, b ; aw, &8).

Similarly, starting from abc (where a, b are not necessarily the
same as above), and proceeding as before, we can construct a

202 THE THEORY OF RELATIVITY

general right-handed bivector, R, consisting of two ordinary, real
vectors R15 R2. This will be transformed by Q[ ] Qc, i,e. so that

R' = <2R<2o, (38)

and will, therefore, have again the two real invariants

R?-R? and (R^). (39)

Both L and R can be used, with equal convenience, for relativistic
purposes, and will be found useful for the treatment of electro-
magnetic questions.

To illustrate the above properties by a simple kinematical example,
take, as the generating factors, the velocity- and the acceleration-
quaternions of a particle. Then

i.e., after a slight calculation, in terms of the ordinary velocity p and
acceleration a;

= -q/3a.

Thus, besides (LjL2) which vanishes, obviously, we have the
invariant (Z12-Z22 and, therefore, also)

-c JZ*=L* - af s/i-£2sin'(p, a),

and this invariant has a simple kinematical meaning. For it is
identical with the absolute value of the rest-acceleration a of the
particle, as given by (14).

Returning to our general topic, let us consider the product of any
number of left-handed bivectors. Then we shall see, by (33), that, in
transforming it, all the internal Q's and <2c's, as it were, neutralize
one another (QQC=I)> and what is left is only the Qc at the
beginning and the Q at the end of the whole chain, exactly as for a
single L. In other words, the vector part of the product of any number
of left-handed bivectors is again a left-handed bivector. Similarly,
we see, by (38), that the vector part of the product of right-handed
bivectors is again a right-handed bivector. But we shall hardly find a
physical application of such products.

What will turn out to be rather important for such application
is the product of one of the original physical quaternions into a
bivector. Of such a nature will be the ponderomotive force in an
electromagnetic field.

PHYSICAL QUATERNIONS 203

Notice, therefore, that if a be any physical quaternion covariant
with q (not necessarily that already involved in L or R), the product
tfL will transform into

fl'L'= QaQQcLQ= QaLQ,

that is to say, <zL will again be covariant with q. So also will "Ra
be covariant with q. And similarly will I*ac and ac"R be covariant
with qc. In short symbols,

tfL ~ Ra - ?, (40)

Each of these products can be, and is, in fact, used for relativistic
purposes. As regards their structure, they are biquaternions, in
Hamilton's (not in Clifford's) sense of the word, that is to say,
quaternions, of which both the scalar and the vector parts are com-
plex.* But, as we shall see in the next chapter, any one of such
biquaternions can be split into a pair of our original physical
quaternions, each ~q or ~qc in the case of (40) or (400) respec-
tively. In this way we fall back to the quaternions considered at
the outset.

Thus, the product of any number of antivariant physical qua-
ternions

...abcdec...

will furnish us (after the rejection of the invariant scalar part) a
bivector L or R, which is transformed by Qc [ ] Q and Q [ ]QC
respectively, or again, biquaternions consisting of pairs of primary
physical quaternions, which are transformed by Q [ ] Q, or by

<2c[ ]<2..

And, as was already remarked, products of covariant factors, such
as ab, are out of question.

As concerns the operation of division by a physical quaternion,
we know that it is reduced to multiplication by its reciprocal. Thus,
it will be enough to observe that the reciprocal of a physical
quaternion is again a physical quaternion. For we have

Thus, for example, if a = ia + A, and L^I^-

204 THE THEORY OF RELATIVITY

and the tensor Ta is a relativistic invariant. Notice that a and a~l
are mutually antivariant.

Finally, notice that any one of the above factors may be replaced
by the quaternionic differential operator

or by its conjugate Dc, which is ~qc. Thus, for example, if the
quaternion 3? ~ q be a function of time and the coordinates, then
V£>c& will be a left-handed bivector; and so also will VZ><£C be a
right-handed bivector. For, independently of their differentiating
power, these operators behave with respect to the Lorentz trans-
formation exactly as any of our primary quaternionic magnitudes.

CHAPTER VIII.

FUNDAMENTAL ELECTROMAGNETIC EQUATIONS.

IN this chapter we shall consider, from the relativistic standpoint,
the fundamental, or 'microscopic,' equations of the electron theory
and their consequences. These equations, written in their ordinary
vector form, are, as under (i.) and (u.), Chapter II.,

^T + PP = c - curl M ; p = div E

-~r = - c . curl E ; div M = o

and

Here, p is the velocity of a charge-element with respect to that
framework S, for which, to begin with, the equations are supposed
to be rigorously valid ; P is the ponderomotive force, per unit
volume, and Jf the ponderomotive force per unit charge, or the
electric force.

First of all, we have to ask whether these equations satisfy the
principle of relativity, that is to say, whether they preserve their
form when we pass from the system S(t, x,y, z) to another system
S'(t', x,y, z) moving with uniform velocity relatively to S. And
if -.the answer be, as it is in fact, in the affirmative, what are the
connexions between E', M', the dielectric displacement and the
magnetic force as estimated from the ^"-standpoint, and these field-
vectors as estimated by the ^-observers ? To answer both of these
questions, first with regard to the differential equations (i.), we could
follow the way originally taken by Einstein, viz. subject the time
and the coordinates involved in the differential operators to the

206 THE THEORY OF RELATIVITY

Lorentz transformation x = yv(x + #/'), etc., and expressing p in terms
of p' by means of his addition theorem of velocities, show the in-
variance of the form of these equations, and finally gather together
the terms which in the transformed equations play the part of the
field- vectors.* But the shortest method to obtain these results is
to write the four equations (i.) in their condensed quaternionic
form,

C, (i)

as given in Chap. II., and to test the constituents of this equation
with regard to their relativistic qualities.

Here, it will be remembered, B = M-fE, while

C=p[t+±j>l (2)

or, in terms of the velocity-quaternion, (3^), Chap. VII.,

C=£Y, (2a)

cyP

where yp = (i -/2/^2)~^

Keeping this in mind, consider the equation (i). We know
already that the differentiator D behaves exactly as a physical
quaternion, viz. that D~q. The only thing, therefore, we still
require, is to find the nature of the current-quaternion C.

Now, the electric charge de of any individual portion of an electron
is a relativistic invariant, i.e. if dS be the volume of that portion,
and dS' its ^"-correspondent, then

pdS=P'dS'. (3)

In fact, taking the divergence of the first of (i.), we have

o = ^ + div (pp) = ^ + (PV) p + p div p,

which is known as 'the equation of continuity,' or, denoting by —
the rate of individual change, as on p. 31,

dp

* An outline of this way of treatment, which may be helpful to some readers,
will be found in Note 1 at the end of the chapter.

CHARGE AND CURRENT 207

whence, multiplying by dS and observing that -,(dS} = dS. divp,*

Thus, the charge, as estimated from the ^-standpoint, is invariable
in time, notwithstanding the motion and deformation of the volume-
element we are watching. This being the case, we can imagine the
charge first fixed in 6" and then set it into motion, bringing it by and
by to the velocity v, when it will be at rest in S'. Claiming, there-
fore, in the name of the principle of relativity, the same rights for S'
as for S, we shall have de = de. (If the reader does not like this
kind of proof, he can simply postulate the invariance of charge, and
verify a posteriori, after having obtained E' in terms of E, M, that
this postulate is fulfilled.)

On the other hand, remembering that volumes are transformed
in the same way as longitudinal dimensions, and denoting for the
moment by dS§ the rest-volume of the element considered, we
shall have

and dS' =
or

ypdS=yp,dS'.
Therefore, by (3),

that is to say, p/yp, the coefficient of Y in (20), is an invariant.

Now, as we know from the last chapter, Y is a physical quaternion.
Therefore, C, the current-quaternion, as it was already called in
Chapter II., is again a physical quaternion, like the standard q,

C*q,

as well as D ^q.

This proves the invariance of the form of the equation (i), or
of the equations (i.), with respect to the Lorentz transformation, and
gives at the same time the connexion between B' and B.

In fact, since C = QCQ, we have from (i)

*Cf. my Vectorial Mechanics, p. 126.

2o8 THE THEORY OF RELATIVITY

and inserting QQC = i between D and B,

where E' = QCEQ* Thus, B, the electromagnetic bivector, is a
left-handed bivector.

Or, to obtain this bivector in its typical form Vacfr, we may
proceed as follows. Operate on both sides of (i) with DG. Then

But DCD is an invariant. This, therefore, is already the required
form. We need not even put the prefix V before D^C, since
SDcC=o, as we shall see when we next return to the last equation.

Thus, B is a left-handed bivector, having the same structure and
the same transformational properties as our L of the last chapter.
Henceforth we can consider it as the standard of physical bivectors,
in the same way as q has been the standard of physical quaternions.

It will be found convenient for subsequent work to write throughout
L (instead of our previous B) for the electromagnetic bivector, f thus:

L = M-*E. (4)

The quaternionic equivalent of the electromagnetic differential equations
(i.) will now be

and the transformation formula of the electromagnetic bivector

(5)

The in variance of the formula (n.) for the ponderomotive force
will, with equal ease, be proved later on. Meanwhile let us fix our
attention upon (5).

As already pointed out in the last chapter, Q and Qc being
unit quaternions, the square of the electromagnetic bivector is an
invariant, i.e.

*That the product Qc^Q is, in fact, a pure vector (i.e. a scalarless quaternion),
like B, we see at a glance. For the conjugate of QC"BQ is QCBCQ= - Qc^Q, so
that the sum of that product and of its conjugate is nil. Q.E.D.

tAnd correspondingly, in what follows, R for the complementary bivector
M + iE, which will turn out to be right-handed.

ELECTROMAGNETIC BIVECTOR 209

Now, by (4),

and similarly for L'2. Thus we have the two real invariants

±(M2-&) and (EM). (6)

The first of these invariants, the difference of the densities of the
magnetic and the electric energies, is the electromagnetic Lagrangian
function per unit volume.* The second invariant, the scalar product
of E and M, has no particular name of its own. Notice that what
is called a pure electromagnetic wave is defined by J/2 = E? and
(EM) = o. In words : energy half electric and half magnetic, and
E and M perpendicular to one another. Using the electromagnetic
bivector we can characterize pure waves more shortly by L2 = LL = o.
At the same time we see that a wave which is pure from the .S-stand-
point is equally pure from the .S'-point of view. In short, purity, at
least in this domain of relations, is an invariant property. But this
only by the way.

Next, to develop (5) into its vectorial form, remember that, by (50),
Chap. V.,

to . to

(9 = cos- + u . sm -5

2 2

where u is the unit of v, the velocity of S' relative to S, and where w
is the imaginary angle previously defined. Multiply out the right
side of (5). Then

L' = ( i - cos w) . u(uL) + cos w . L + sin to . VLu.

From this intermediate form we can easily see that L' is obtained
from L by turning it about U, the axis of the quaternion Q, through w,
the double of the angle of that quaternion. Such then is the office
of the operator Qc [ ]Q. This is only a particular instance of
a theorem of the calculus of quaternions, given by Hamilton
himself, t

* The properties of this function, belonging to the elements of the Electron
Theory, are given in Note 2.

f If k be any quaternion, k~^ its reciprocal, and x any quaternion to be operated
on, then the operator <£-1 [ ] k turns the vector of x about the axis of k through
double the angle of k, while the scalar (s) of x remains unchanged, of course
(since k~lsk = sk~lk = s). Cf. Tait's Quaternions, 1890, p. 75.
S.R. O

2io THE THEORY OF RELATIVITY

But let us write the last formula in terms of y, which is an
abbreviation for yv = ( i - /32)~ -', ft = vjc. Remembering that cos w = y
and sin<o = t/3y, we have

L' = ( i - y) u(uL) + yL + - yVLv,
or, employing again the longitudinal stretcher c, of ratio y,

L' = y[^L + ^VLv], (7)

and splitting into the real and the imaginary parts, according to (4),

Or, finally, in Cartesian expansion, using Ii2>3 for the rectangular
components of the vectors taken along the direction of motion and
perpendicular to it (right-handed system of axes),

+ RM.\ \

(7*)

These are the relativistic formulae for the transformation of the
electric and the magnetic vectors, as obtained by Einstein. They
agree entirely with those given by Lorentz in his modified theory
(see p. 86). Notice that, in passing from the S- to the ^"-standpoint,
the longitudinal components of E, M remain unchanged, while the
changes brought about in their transversal components involve the
vector products VvM and VEv and the coefficient y.

Multiplying both sides of (5) by Q as a pre factor and by Qc as a
postfactor, we have at once

But Qc follows from Q, and vice versa, by a mere change of the sign
of v. Thus, the inverse transformation, giving E, M in terms of
E', M', is obtained by changing the sign of v in the vector formulae,
or by writing - p instead of ^ in their Cartesian expansions, and by
transferring the dashes, to wit

), etc.,

CONVECTIVE FIELDS 211

as the reader may also prove by solving (7^). This shows once
more that none of the systems of reference is privileged.

The invariance of electric charge, used at the outset, can now be
directly verified by differentiation of the transformed electric vector
or of its components.*

The applicability of the above formulae of .transformation is
obvious. For, if we know a solution of the electromagnetic differ-
ential equations for one of the legitimate systems of reference, we
can deduce from it at once the solution for any other of such
systems. Now, the problem of integration may be much easier for
one of these systems than for any other, owing to some particular
simplicity of the conditions as stated from the standpoint of the
former system. Whence the advantage of the method.!

The simplest solution of the electromagnetic equations is an
electrostatic field corresponding to a given distribution of charges
(electrons), which are all fixed with respect to a legitimate frame-
work, say S'. The ^-correspondent of this will be the electro-
magnetic field accompanying a system of electrons in uniform
translational motion, with velocity v relative to S, or what is called a
convective field. The framework S' will be the rest-system belonging
permanently to these charges. It will be good, before proceeding
further with our general subject, to consider this example at some
length.

Let us suppose, therefore, that we have in S' a purely electrostatic
field, so that E' = - V'<£', where <£' is the scalar potential of the given
distribution of charge, while M' = o. Then, remembering that the
inverse of the first of (7 a) is

we shall have, from the Spoint of view,
i.e., in Cartesians,

*See Note 3.

fThe reader will find it useful to compare this procedure carefully with
that contained in Lorentz's ' Theorem of corresponding states,' as given in
Chapter III.

212 THE THEORY OF RELATIVITY

The second of (70.) gives us at once M in terms of E, viz.

M = -VvE=TVvE,

c c

since the stretcher e acts along v, while the vector product is normal
to v.

Thus we have for the most general convective field, accompanying
any system of charges which moves as a whole with the uniform
translational velocity v relative to S,

Here E' = - V'<£', the scalar function </>' being the electrostatic potential
of the given distribution of charge fixed in S'. The problem is
therefore reduced to finding, for each particular case of distribution,
the scalar potential <£'. Observe that this is the potential of E',
while E has no such potential. Notice, further, that the magnetic
lines, due to the motion of charges, are everywhere normal to both
E and the direction of motion. And since E' is coplanar with E, v,
the magnetic lines are also at right angles to E'.

The gradient or slope V'</>' can easily be replaced by V<£'. In
fact, measuring x along the direction of motion, so that x = y(x - vt\
and remembering that, by assumption, 3<£'/3/' = o, we have

M- M M_M M_M

?)x *dx' 'by 'by' *dz *dzf
i.e.

€V'<£' = V<£',

so that the first of (8) can be written

Thus, the displacement E, as already remarked, has no scalar
potential. But the electric force Jf, or the ponderomotive force per
unit of charge carried along with 5', has such a potential, exactly
as in Lorentz's treatment, given in Chapter III. p. 81. In fact,
remembering that in the present case p = v, we have by (n.) and
by the second of (8),

= E + ^ VvVvE = E - /32[E - u(uE)],

CONVECTIVE FIELDS 213

or

and by our last formula,

(9)

Thus <£'/y is the scalar potential of the electric force. This is the
convection potential ®{ Chap. III., the above equation being identical
with formula (21) of that chapter, in which <£ was y<f>'. The same
result may be deduced more directly from the transformational
properties of the ponderomotive force, to be developed later on.

Since y is constant throughout S', the surfaces of constant con-
vection potential and those of constant <£' overlap. We see, therefore,
that the lines of electric force Jf (but not those of displacement E)
cut perpendicularly the surfaces of constant electrostatic potential
of the rest-system, <£' = const. The electric force and displacement
of that system are identical, of course, i.e. Jf = E'.

To illustrate the general formulae (8) of the convective field,
suppose that the distribution of electric charge in S' is in homogeneous
concentric spherical sheets round O', the origin of the coordinates
or the origin of the vectors r'. Then <j>', and consequently also E ',
will be functions of r alone, and the lines of displacement in S'
will be straight and radial or, say,

E'=/(r).r', (10')

where f is a scalar function of its argument. By the fundamental
formulae of transformation, r' = er — vy/. Now, since the whole field,
together with the charges, moves past 5 without being deformed, it
is enough to consider it at one single instant. Let this be the
instant t = o, when O coincides with O, the origin of the ^-coordinates
or of all vectors r. Then

r' = er,

and, by (8),

E = y/(r).r }

! («)

M = iy/(r').Vvr,

so that the dielectric displacement in S is again in straight radial
lines, while the magnetic lines are circles normal to the direction
of motion and centred upon the axis of symmetry passing through O.

214 THE THEORY OF RELATIVITY

The whole electromagnetic field is symmetrical, of course, round
this longitudinal axis. Since r = €~1r', or

the spheres r = const, become, in S, oblate ellipsoids of revolution,
as in the FitzGerald-Lorentz contraction, i.e. having

for their semiaxes. These are known as Heaviside ellipsoids. Such
then will be the surfaces of constant convection potential, and the
lines of electric force (Jf), cutting these ellipsoids at right angles,
will be parabolic arcs, contained in the meridian planes.

If s = (y2 + z2)^ be the distance of a point from the axis of
symmetry, we have

or also, denoting by 0 the angle contained between r and the axis,

r' = yr*Ji-/32sm20. (n)

This is to be substituted in each particular case for the argument
of the given function f in (10).

Take, as the simplest case of the above kind, a single sphere of
homogeneous surface-charge, or a Lorentz electron. Call its rest-
radius J? and its total charge e (which, as we know, is the same
thing as e'). Then E' — o inside the sphere r' = ^?, and consequently
also J£ = Q inside the oblate ellipsoid y2 x2 + s2 = J?2, while at the
surface of and outside the electron*

E' = -eT'

• 4
and therefore

that is, by (n),

with the magnetic force M = - VvE to match. For any given 6 the

value of £9 and consequently also that of M, are inversely propor-

*In Heaviside's rational units.

ELECTROMAGNETIC MASSES 215

tional to the square of the distance from the centre of the electron.
The unit tubes of displacement, though everywhere radial, are
crowded towards the equator, and the more so, the greater the
velocity of motion. At any given distance r, the density of the
tubes at the equator is greater than that at the poles (0 = o or TT)
in the ratio Ev& :E0=i:(i- /!-)%.

From the above, widely known, formulae the longitudinal and the
transversal electromagnetic masses of the electron may be easily
deduced in the usual way. The flux of energy or the Poynting
vector being

= VEVvE = & v - (Ev) E,

we have for the electromagnetic momentum, per unit volume, by
(30), Chap. II.,

where u is the unit of v and £1 the longitudinal component of E.
Integrating through the whole field (from r=R till r=oo) and
taking advantage of its axial symmetry, we obtain, for the total
electromagnetic momentum*

o

yv, (13)

whence the longitudinal electromagnetic mass ml of the electron and
the transversal one, mt, defined by mi = dGjdv, mt=Gjv:

Wj = w0y3, mt = mQyt (14)

where

These are the well-known formulae of Lorentz, as mentioned
previously. They are valid for an electron of homogeneous surface-
charge. In the case of volume-charge, we should obtain for the
electromagnetic momentum -| of the above value, so that (14) would
continue to hold with m0 equal to -| of the above,

* See Note 4 at the end of the chapter.

216 THE THEORY OF RELATIVITY

The electromagnetic momentum can, in either case, be written

G = w0yv. (16)

Thus, #20, the electromagnetic rest-mass, plays the same part as the
rest-mass, of any origin, in the relativistic dynamics of a particle.
Cf. (24), Chap. VII.

Having for the present sufficiently illustrated the transformational
properties of the electromagnetic bivector, let us now return to our
general subject.

Consider again the equation

C (i.a)

embodying in itself the whole of the electronic differential equations
(i.), and showing at the same time their invariance. Operate upon
both of its sides with Dc. Then

But DCD is the Dalembertian,
c)2 &

and this is a purely scalar operator; that is to say, if applied to a
scalar it gives a scalar, and if applied to a vector it gives again a
vector. Now, L is scalarless. Therefore

SAC=o. (17)

This is the equation of continuity. In fact, its developed form is,

by (2),

But this only by the way.

Next, introduce an auxiliary quaternion <£, satisfying the differ-
ential equation

-C (18)

and the supplementary condition

Then, when <£ is found, for any prescribed C, the electromagnetic
bivector will be given by

L=-A*- (20)

POTENTIAL-QUATERNION 217

Now, Z>cZ> = n, being the norm of D~q, will be an invariant,
as was already remarked on p. 113. Therefore, by (18), <£ will
be a physical quaternion, having an imaginary scalar and a real
vector. Write it, therefore,

3> = f<£ + A~^, (21)

and call it the potential-quaternion, since the whole electromagnetic
bivector is derived from it by simple differentiation. The corre-
sponding world-vector is called the four-potential.

The scalar part of * is i times the usual scalar potential, and its
vector part is the vector potential. In fact, splitting (20) into the
real and the imaginary parts, we obtain at once

curlA,

P ™

Bas-v*

while the condition (19) becomes

and these are the familiar formulae of the electron theory, as
employed incidentally in Chapter III., p. So. The differential
equation (18) splits, of course, into the familiar pair of equations,

identical with (16), Chap. III.

According to (21), <£ and A are transformed as ct and r. Thus,
for instance, if we have in S' a purely electrostatic field, i.e. if A' = ot
then, for the convective field, as estimated from the ^-standpoint,

as mentioned above, and

as in (19), Chap. III.

So much as regards the potential-quaternion and its relationship
to the electromagnetic bivector.

Next, observe that instead of the above L = M - L E we might
equally well have taken the complex vector

, (22)

2i8 THE THEORY OF RELATIVITY

which can be called the complementary electromagnetic bivector.
Then we would have obtained as the condensed equivalent of the
fundamental equations (i.), instead of and in exactly the same way
as (1.0),

AR=CC5 M

where Cc is the conjugate current-quaternion p(i-p/c). Operate
on both sides of this equation with D. Then the result will be
DR = Z>6'C. And since the Dalembertian is an invariant, we see
at once that R is a right-handed physical bivector,* i.e. that

•R'=QKQC. (23)

Henceforth R can be considered as the standard of all such bivectors,
just as L became the standard of the left-handed ones. Obviously,
the differential equation (i.£) is invariant with respect to the Lorentz
transformation, i.e.

A'R' = C.

(i.a) and (i.£) differ, of course, only formally from one another;
each, when split, gives the four electromagnetic differential equations
(i.). Thus, as far as the equations of the field and all their con-
sequences are concerned, we do not need both L and R, but
require only one of them at a time.

For some other purposes, however, the simultaneous use of both
bivectors will prove to be very advantageous.

Their symbols, being the initials of 'left' and 'right,' are chosen
so as to remind the reader of their transformational properties.
In connexion with these, L can admit a physical quaternion, co-
variant with ^, only on its left as neighbour, and R only on its
right. And vice versa, if the neighbour is covariant with qc.

Now for the outstanding proof of the invariance of the funda-
mental formula (n.) for the ponderomotive force. To obtain this proof
we have only to write that formula in terms of legitimate relativistic
magnitudes.

If we multiply our left-handed electromagnetic bivector, on the
left side, by any physical quaternion -^, then, as in (40), Chap. VII.,

* This property of R = M + iE may also be deduced directly from that of L = M - tE.
For it is easily proved that if (for any pair of real vectors A, B)

A - tB is a left-handed physical bivector,

then A + iB is a right-handed physical bivector,

and vice versa. (See Note 5.) This simple theorem will be found useful later on.

PONDEROMOTIVE FORCE 219

the resulting product will again be transformed like q. Now, the
current-quaternion C being precisely such a quaternion, consider the
full product

CL.

This then will again be transformed by Q[ ]Q. Develop it, by (2)
and (4). Then the result will be

Cl = J?+iFMt (24)

where

(24*)

and Fm, the magnetic analogue* of this,
F =

Now, the vector part of F is exactly P, the ponderomotive force per
unit volume, as given by (n.), and the scalar part of F is i/c times
the activity of this force. Thus,

^(Pp) + P. (25)

Observe that the whole product CL, though covariant with the
standard <?, has not the structure of ^, since it is a full biquaternion,
in the Hamiltonian sense of the word. But F, and its magnetic
analogue, have each the structure of ^, i.e. a real vector and an
imaginary scalar.

Similarly, the complementary B being a right-handed bivector,
multiply it on the right side by C. Then the product "RC will again
be transformed by Q[ ]Q. Develop it. Then, by (2) and (22),

(24a)

with precisely the same meanings of F and Fm as above. This
again is a full biquaternion.

Now, since both biquaternions, CL and "RC are transformed by
Q[ ](?, this will also be the relativistic transformer of their sum and
of their difference. Leave alone the sum, which would give the

* The reader will have remarked that in this and in all other cases the magnetic
analogue is obtained from the electric original, and vice versa, by writing M for E,
and - E for M. In the present case, Fm has, as far as we know, no immediate
physical meaning. And since we shall need F only, it seemed convenient to leave
it without the subscript e.

220 THE THEORY OF RELATIVITY

physically uninteresting JFm, and take half the difference of (24)
and (24^). This will give

^=|[CL-RC]. (n.a)

Thus, we see that F taken by itself (as well as Fm} is co variant
with q. And since F has also the structure of q, it is a physical
quaternion, and may as such be called the force-quaternion per unit
volume. It has a dynamic vector, the ponderomotive force per unit
volume, and an energetic scalar, proportional to the activity of
that force.

At the same time we have obtained for F the expression (ii.a),
and we know that the vector part of this is equal to P as given
by (ii.). Now, (n.fl) transforms into

and the vector part of this quaternion is again

which proves explicitly the in variance of the formula (n.) with respect
to the Lorentz transformation.

Thus, the whole of 'the fundamental equations for the vacuum/
as (i.) and (n.) are called, satisfy rigorously the principle of relativity,
and it was for this reason possible to incorporate them entirely in
the new doctrine.

By (25) we have, identically,

S^Cc = o, (26)

and therefore also, by (20),

S^Kc = o. (27)

In four-dimensional language we should say that the four-force,
equivalent to the quaternion Ft is perpendicular to the four-
current, and consequently also to the world-line of the element of
electric charge acted upon. We met with this property when
treating the dynamics of a particle moving under the action of a
force of any nature whatever. See (21), Chapter VII.

Remember that what is, in our present case of electromagnetic
action, a physical quaternion is the force-quaternion /^per unit volume.
That is to say, what is transformed as r (and ct) is P, the pondero-
motive force per unit volume (and c~l times its activity), and not the
total force acting upon an electron or upon its volume-element.

POXDEROMOTIVE FORCES 221

The latter is not the vector part of a physical quaternion. But, on
the other hand, we know that

yp x volume
is an invariant. Therefore

yp x vol. x F~ q,

that is to say, yp times the force-quaternion calculated for any particle
of electricity is again a physical quaternion. Such then is the trans-
formational property of ponderomotive forces due to an electro-
magnetic field.

Now, if one of these forces is in equilibrium with a force of
any other origin, from the standpoint of the system S' (so that
the particle acted upon is at rest with respect to that system), then
these two forces have also to balance each other when estimated
from the standpoint of any other legitimate system S. For relatively
to S, the particle in question will move uniformly. Hence the re-
quirement, that ponderomotive forces of any origin shall be transformed
in exactly the same way as those of electromagnetic origin* i.e. so that

yp [total force + - times its activity] ~ r + ict.

Here * total force' means the force acting upon a particle whose
velocity relative to S is p, or upon a body of any dimensions if all
its parts happen to have the same velocity.

Now, what in Chap. VII. has been called the Newtonian force,
N, satisfies exactly this relativistic requirement. In fact, according
to the formula (18) of that chapter (where y stands for yp),

r,(N + ,„) = *
is a physical quaternion, and, as we have seen, v=-(Np). It is

precisely for this reason that the Newtonian force, not the Min-
kowskian, has been considered as the force, and the magnitude
mc-(y — i), whose rate of change has been equal to (Np), as the
(kinetic) energy of the particle.

This procedure of transferring the transformational properties from
certain physical magnitudes to others of the same kind is an
important feature of the theory of relativity.

* This fixes, of course, only the transformational properties of forces of any kind,
without obliging us, however, to attribute to all such forces a common electro-
magnetic origin.

222 THE THEORY OF RELATIVITY

After this short digression of a general nature, let us return to
our electromagnetic topic.

The formula (ii.a), obtained above for the force-quaternion F,
has nothing to do with the differential equations (i.) of the electro-
magnetic field. It is only another form of the original formula (n.)
for the ponderomotive force. Now, use those differential equations
in their quaternionic condensation (i.a), that is to say, substitute
C=D"L. Then the double of the force-quaternion will be

2^=Z>L.L-R.Z>L, (28)

where the dot stands for a separator, stopping the differentiating
action of D. This formula, when subjected to a slight, though
somewhat peculiar change, will prove to be very convenient for
further application. The peculiarity of the formal change alluded
to, consists in this, that it requires us to give up an old habit.
Hitherto, in conformity with the general convention, we have always
used the differential operator D as a 'prefactor,' i.e. acting forward
only, just as an ordinary scalar differentiator, such as B/9/, is used.
Now, the position of a scalar being a matter of indifference, it
would be utterly useless and extravagant to write 3/3/, for instance,
behind the scalar or vector function to be differentiated ; for such

expressions would mean just the same as ^- or ~-. But the case is

different when the differentiator has the nature of a vector, as the
Hamiltonian V, or of a quaternion, as D. Since the multiplication
of vectors, and more generally of quaternions, is non-commutative,
we obviously deprive ourselves of possible advantages if we limit
the rdle of quaternionic differential (or other) operators to that of
prefactors. Henceforth, therefore, we shall use D as an operator
acting both forward and backward* i.e. as both a prefactor and a
postfactor, and we shall, for instance, write

B[Z>]L = RZ>.L + R.Z>L, (29)

where the dots stop Z>'s differentiating power, and where the brackets
(which. could also be omitted) are used for better emphasizing the

* To cut short any justification of this departure from convention we could repeat
here Oliver Heaviside's words, who, in a similar situation, says simply : ' A cart
may be pulled or pushed.' Then, as regards non-differential operators, we have
learned long ago from J. W. Gibbs to employ linear vectoijt as both post/actors
and prefactors.

PONDEROMOTIVE FORCE 223

bilateral action of the enclosed operator. The only thing to be still
explained in this symbolism is the meaning of RZ>, which is unusual
inasmuch as the operator D follows the operand. Now, if D were an
ordinary quaternion, that is a quaternionic magnitude, with s, v as
its scalar and vector parts, we should have, by elementary rules,

RZ> = Us + VRv - (Rv) = jR - YvR - (vR).

Writing therefore 3/3/ instead of s and V instead of v, the plain
meaning of "RD will be

-divR.

This settles the question. Notice that Z>R could not be used for
relativistic purposes, since R is right-handed.

Now, to see the utility of RZ>, return to (i. <£), by which Z>CR = Cc .
Take the conjugate of each side, and remember that Rc= -R.
Then, by the rule of conjugate of a product,

and consequently, by (i.a),

Z>L= -RZ>,
and, substituting this in (28),

2F= -RZ>.L-R.Z>L= -R[Z>]L.

In this way we obtain the required short expression for the force-
quaternion, in terms of the electromagnetic bivectors,

^=-|R[Z>]L. (ii. £)

Thus, R[ ]L, when applied to Z>, or more correctly, when exposed
to the bilateral differentiating action of D, gives the force-quaternion.
We shall see in the next chapter that the same operator R[ ]L,
when applied to an ordinary vector, e.g. the normal of a surface-
element, will give us the corresponding stress, and, when applied
to a scalar, the density and the flux of electromagnetic energy.

As regards the matrix-equivalents of our bivectors and quaternionic
equations, it seemed preferable, for the sake of avoiding any possible
confusion, not to insert them in the text of this chapter. Some of these
equivalents are given in Note 6, which, together with our previous remarks
on matrices (Chap. V.), will perhaps be found sufficient.

224 THE THEORY OF RELATIVITY

NOTES ON CHAPTER VIII.

Note 1 (to page 206). Take first the case p = o, that is to say, consider
the equations (i.) outside the charges. Measure x along v, the velocity
of S' relative to S. Then

_3_= _V__n J3 3 3 „ |3 33 33
and the equations

I ^^1 _ ^"3 _ U2™2 ta\

and

div E = -~-* + -TT-^
o^r qy

will be transformed into

and

Take the sum of the first and (3 times the second of these equations.
Then the result will be

Thus the form of the equation (a) reappears. Treat similarly the remain-
ing of the equations contained in (i.). Then the whole of these equations,
with /a=o, will reappear in dashed letters, thus :

_ _ _ _ _

<~\ .. — f^\ i y™^ /

r 3/ 3y 02
where

the common factor ^r(v) being thus far an indeterminate function of ?/,
Avhich for v = o reduces to unity. But solving the last six equations with
respect to the non-dashed components and claiming mutually equal rights
for the two systems, S and S', we obtain at once

and, for reasons of symmetry,

i
so that

LAGRANGIAN FUNCTION 225

and these are the required formulae of transformation, identical with (jb)
of this chapter.

Xext, pass to the general case of divE = /o^o. Bring in the omitted
terms />/,, etc., the components of /op, and, by means of the addition
theorem of velocities, express p in terms of p' and V. Then the whole of
the general equations collected under (i.) will reappear in dashed letters,
thus :

where

or

and where the components of E', M' are still connected with those of
E, M by the above formulae (b). The details of calculation, similar to
those for p = o, may be left as an exercise for the reader. By working it
out fully he will convince himself best of the advantages of shortness and
simplicity offered by the quaternionic method employed for the same
purposes in the text of the chapter.

Note 2 (to page 209). The difference of the magnetic energy Um and
the electric energy Ut,

L=Um-U. = i J ( J/2 -

has been called the Lagrangian function, because it has been remarked
that the fundamental electronic equations, (i.) and (li.), can be condensed
into a single variation-formula having the structure of Hamilton's

Principle (or the principle 'of least action'), 8 / 2...=o, in which precisely

Jti

that difference of the two kinds of energy appears (along with other
possible terms) under the sign of integration. This result is hardly more
than a purely formal condensation of the original equations. And
since some authors have attributed to it an exaggerated mechanical or
dynamical significance, it may be well to give here a short sketch of the
bare result and of the method by which it is usually obtained.

Consider a region of space, bounded by the surface <r, fixed in that
system S in which the equations (i.) and (u.) hold. Let p = o at each of
the points of the surface <r, whose choice is otherwise arbitrary. Let the
space region, whose volume-elements will be denoted by dS, contain any
system of electrons, or more generally, of charges which may be either
free or imprisoned in particles of matter in the ordinary sense of the
word. Let the * virtual displacement ' consist of a space displacement Sr
of matter and electrons and of a local variation 8E of the electric vector.
Let £r and 8E be such continuous functions of time and space, as leave
s.R. p

226 THE THEORY OF RELATIVITY

the charge of each element of matter unchanged. With this assumption,
and since p = divE, the distribution of the infinitesimal vector

(b)

will be solenoidal, i.e. such that div(S'C) = o. Let IV\>& the infinitesimal
virtual work of the ponderomotive forces of electromagnetic origin only,
i.e. by (IL),

W= (P8r)<tS= //)(3r[E + -VpM])^.

J J C

Then, by the differential electronic equations (i.), and after a long but
easy calculation (the details of which, together with the literature of the
subject, will be found in Lorentz's article in the Rncyklop. der mathe-
matischen Wissenschaften, Vol. V2, pp. 167 et seq. ; Leipsic, 1904):

- Ue) - <$ Um) - (Xn)flT(r, (c)

where X is the infinitesimal vector

X = VA SM - VA + VE S'M,

A being the usual vector potential, so that M = curl A. The symbol 8r
denotes the variation which would correspond to a change of the total
electric current

by! S'C, the elements of matter being kept fixed. This amounts to
defining S'M by c . curl S'M = S'C, so that

8'C/m= J(MS'MXS= | (S'M. curl A)</S
= |(A curl S'MKS+ J(nVAS'MXo-

Such then is the value of the variation appearing in the second term
of (c). But this only by the way.

Now, let a- expand indefinitely. Then, in virtue of the usual assumption
as to the behaviour of the field ' at infinity,' the surface integral in (c} will
vanish, and

- Ue) - (8' Um). (d)

On the other hand, if T be the usual kinetic energy of matter and V the
potential energy, corresponding to the forces of non-electromagnetic
origin (which are supposed to be conservative), we have, by d'Alembert's

LAGRANGIAN FUNCTION 227

principle applied to the ordinary, non-electromagnetic masses m of the
system (if there be any such masses),

Any relativistic amendment of d'Alembert's principle is here disregarded,
of course. Combining the last two equations, integrating from /=/t to
/=/2, and assuming that Sr and SE vanish at these limiting time instants,
we obtain, finally,

(e)

that is to say, Hamilton's Principle, in which to the ordinary kinetic
energy the magnetic energy Um and to the potential energy the electric
energy Ue is added. In particular, if the whole energy is electromagnetic,
as in Abraham's theory, we have simply

8

The more general equation (e) corresponds to the broader view held
by Lorentz.

Thus, L—Um-Ut plays the role of the Lagrangian function. Con-
versely, assuming 3£y9/+/>p = r.curlM, with /a = divE, and divM=o,
the remaining fundamental electronic equations, i.e.

and P=/o[E + -VpM],

can be deduced from (e). For slowly varying motion of the electrons,
formula (d) gives at once the ponderomotive forces of electromagnetic
origin, corresponding to any set of configurational parameters, in the
well-known Lagrangian form.

Remember that what is invariant with respect to the Lorentz trans-
formation is the Lagrangian function per unit volume, i.e. ^(M2 — E2).
But since ypdS and dfjyp, and consequently also dS . dt are invariant, the
element of * action '

Ldt=(Um-Ue)dt

is an invariant. And so also is the whole 'action' / Ldt invariant with

respect to the Lorentz transformation. It may be noticed here that this
is only a particular instance of a general theorem of relativistic dynamics,
obtained by Planck.

Note 3 (to page 211). Differentiating E^ = E^ Et' =y(JE9~ ftMJ and

Ez=y(Ez+pM^) with respect to .r', y and 2' and passing to x,y, ^, we
obtain the formula (c) of Note 1, in which 7 = 7,.. Thus,

228 THE THEORY OF RELATIVITY

Now, by the addition theorem of velocities (see Chap. VI., and especially
formula (£), p. 169),

whence, by inversion,

Thus p'lypl = plyp, and since yp,dS' = ypdS,

p'dS' = pdS,
which is the required verification of the invariance of electrical charge.

Note 4 (to page 215). Using the formula obtained for g on p. 215, we
have, for the electromagnetic momentum of the whole field,

G = jg dS =

where u is the unit of v and R^ the longitudiual component of E. If E< is
the transversal part of E, the bracketed terms may be written

and since the field is, in the case under consideration, symmetrical
round u, the transversal terms cancel one another in the process of in-
tegration, so that

G = ~ f(E'2 - EfidS= 6u.
For a Lorentz electron of homogeneous surface-charge,

and E = o inside the electron. Writing, therefore, r2-.r2=j2, we have

where the integral is to be taken throughout the 5-space lying outside
the ellipsoid r' ' = (y2^ + y2)- ' = R. But since this ellipsoid is, for the
^'-standpoint, a sphere of radius 7?, it is easier, of course, to perform
the integration in the 6"-space. Thus, remembering that s = s' and
dS=dS'ly (or ihat the functional determinant of A',_y, z with respect to
x>, /, ^ is i/y),

fs2 ,c i /Y2 ,r, i Tsin2^' ,0,

f— ;irtO=— / -^<7O =— / 5T— «O

yjr'6 yj r'4

V 8?r

PHYSICAL BIVECTORS

229

so that

and

which is the required formula.

Note 5 (to page 218). Let A, B be a pair of real vectors and A- iB a
left-handed physical bivector, i.e. such that

This splits into

A' = re. <9CA6> - i . imag. Qc EQ }
and iB' = i . re. QfiQ - imag. QCA.Q, J

where re. and imag. stftnd for 'real part of and 'imaginary part of.'
Now, since Q has a rftfljEgctor Imcf an imaginary scalar, and since Qc is
the conjugate of Q, it is obvious that

mag. =- mag.

and similarly for B. Therefore, by (a),

A' + iB' = re. QAQc + 1 . re. QEQ + imag. QKQC + 1 . imag.

that is to say,

is a right-handed bivector. Q.E.D.

Note 6 (to page 223). Our physical bivector is equivalent to Minkowski's
space-time vector of the second kind and to Sommerfeld's six-vector.
Minkowski represents this world- vector by an ' alternating* matrix

215 °> ^23 »

subjected to the condition that

A being the same matrix as in (36) or (40), Chap. V., and A the
transposed of A. The analogy between "L' = QC'LQ and the last trans-
formation formula is seen at a glance. But the multiplication by a
quaternion is actually less troublesome than the application of a matrix
of 4 x 4 constituents.

230 THE THEORY OF RELATIVITY

The matrix h is built up of six independent constituents (not counting
the diagonal which is always the same). Out of these six constituents
three, not containing the index 4, are real, and the remaining three
imaginary :

7z23, 7/31, 7/12 real,

^u> ^24> ^34 imaginary.

Along with h^ Minkowski uses the corresponding ' dual"* matrix which
he denotes by h*, and which is again an alternating matrix, e.g.

o, 7?34, /z42, 7

^43> °> ^14, ^
^245 ^41 > °> ^1

This is transformed like h. The product of both matrices,

h*h = 7z32 £14 + h^ 7*24 + 7/21 /IM , (a)

which is also the square root of det7z, and

v+v+v+v+v+v w

are invariant with respect to the Lorentz transformation. Both of these
invariants are contained in the square of our physical bivector.
Let, in particular,

Then the matrix h will correspond to the electromagnetic bivector
L = M-iE. (In Sommerfeld's four-dimensional language we should say
that the magnetic components are projections of the six-vector h upon the
planes^, zx, xy, and — i times the electric components the projections
of h upon the planes xl, j/, zl.) With this particular meaning of h the
matrix form of the electronic differential equations (i.) consists of the
equations

the former embodying the first pair and the latter the second pair of the
equations (i.). Here s is the current-matrix,

corresponding to the current-quaternion C*=p(t + p/^). Both of the
equations (d) are contained in our DIt = C. The two invariants (a), (b)
become, in virtue of (<:),

M2-£* and i(EM).

MINKOWSKI'S FORMULAE 231

Both of these are contained in Lr. The ponderomotive force P, (n.), and
its activity are given by the matrix -s/t. In fact, taking the product of
s into /*, by the rule given in the Note to Chap. V., we obtain

-Et, etc., -

i.e.

Since s' = sA, as in (37), p. 143, and h' = AhA, we have s'h'=shA, showing
that the four-dimensional force, per unit volume, is indeed a world-vector

of the first kind. Its quaternionic equivalent is /r=-(Pp) + P, the force-
quaternion of this chapter. The expression "RC-CI* in formula (n.a)
takes the place of the matrix 2s/t.

CHAPTER IX.

ELECTROMAGNETIC STRESS, ENERGY AND MOMENTUM-
EXTENSION TO GENERAL DYNAMICS.

IN the preceding chapter we have seen that the fundamental
electronic equations are invariant with respect to the Lorentz trans-
formation, and we have obtained for the force-quaternion per unit
volume, i.e. for

(i)

the short formula (n.#), p. 223,

^=-pt[Z>]L. (2)

Here D is intended to operate on both R and L, and the only
office of the brackets is to remind us of this bilateral differentiation.

We shall now deduce from this formula the electromagnetic stress
fre together with the density and the flux of energy. All these
magnitudes have already been treated in Chap. II. But now, in
virtue of (2), they will appear in a form which will disclose at once
their transformational properties.

Take first the scalar part of (2). This gives, by (i), and since
SRL= -(EL),

or

(Pp) = - ~- - div $, (3)

where

« = J(RL)

,-KVI* ; (4)

STRESS, ENERGY, ETC. 233

Remembering the meaning of L and R, the reader will see at once
that these are identical with the familiar formulae u = ^(E? + M'2),
•)8=<rVEM. But the above form will better answer our purposes.
Thus, the scalar part of the equation (2) expresses the conservation
of energy, giving the flux of energy or the Poynting vector J, along
with u, the density of electromagnetic energy. Both of these may
also be condensed into the full product,

It is hardly necessary to say that this is not a physical quaternion.*
But the formula recommends itself by its shortness.

Next, consider the vector part of (2). This is, by (i), the pon-
deromotive force,

or, by the second of (4),

P=_i|l-lvR[V]L.
c- 15( 2

Writing V = i9/3,r+j3/3)' + k3/3s;, and remembering that both R
and L are to be differentiated, we have

On the other hand, if / is a stress-operator, i.e. if

is the pressure, per unit area, on a surface element whose unit
normal is n, and if we write in particular, as on p. 48,

then the corresponding resultant force per unit volume will be
or

* R is not the right sort of neighbour for L. In fact, R'L' = QRQfLQ, and
similarly, lW=-

234 THE THEORY OF RELATIVITY

which is exactly of the form of (a). We see, therefore, that

where g = $/<r2, as on p. 51, and where, for i, j, k, and hence also
for any unit vector n,

This is the required formula for the stress. Multiplying out the
right-hand side, the reader will easily obtain

= «n-E(En)-M(Mn),

which is the Maxwellian stress, (20), p. 48. But the above form,
obtained directly from (2), is more appropriate for our purposes.
Again, since the stress is irrotational, or since / is a symmetrical
operator, we have/i = i/J etc., so that we may write, in the last
formula for P,

dx ty -dz /3

where f is to be considered as a dyadic. (See Note i.) Had we
used this short form at the beginning, we might have obtained the
above formula for f even more directly.

Thus, the vector part of the equation (2) gives for the pondero-
motive force the expression

*--g-iK (5)

where g, the electromagnetic momentum per unit volume, and/n, the
stress for any orientation of n, are determined by

(6)

C* 2C

and

/n = |VRnL. (7)

On the other hand, the scalar part of (2) contains the principle
of energy, (3), and gives for the density and the flux of energy the
expressions (4).

In (7) we have the vector part of a product of three vectors.

STRESS, ENERGY, ETC. 235

Now, the scalar part of this ternary product is

SRnL = SEVnL = - (RVnL) = (nVEL),
so that, by (4),

JSBnL^($n).
Consequently, the full product will be

iRnL = ^($n)+/n. (70)

It will be convenient to combine this with (40) into one formula.
Let a- be a real, but otherwise arbitrary scalar, and let us intro-
duce for the moment the auxiliary quaternion

k = to- + n.
Adding to- times (40) to (717), we have

JR£L = ^[($n)-^]+/n~$. (8)

This is valid for any k, that is, for any direction of n and for any
value of CT-.

Since (2) transforms into itself, i.e. into F' = -JR'[Z>']L' for any
legitimate system S', the same thing is true of the equation of
energy (3) and of the formula for the ponderomotive force (5).
Both are invariant with respect to the Lorentz transformation. Thus
we have, in £,

and

P' - _. ?§L _
W

where g' = J'/*2 and where $', u, f are determined by the previous
formulae, i.e. also by (8) with dashed letters. Remembej that f is
the stress-operator in S', so that if n' is a unit vector,Tjf n' =/„, is
the pressure on a unit area whose normal is n'.

What are the connexions between Jp', u\ f on the one side and
J3, u, f on the other side ? To answer this question, return to (8).
Take for k a physical quaternion, so that

k' = to-' + N' = 10-' + JVn' = QkQ,
i.e.

*' = y l> - ~c (vn)]» N' = en - ^yo-v, (9)

236 THE THEORY OF RELATIVITY

n' being the unit of N'. Then R/£L will also be a physical qua-
ternion, ~£r. Denoting, therefore, by (8) the right side of the
equation thus numbered, and by (8') the same expression with
dashes, we have

Writing down Q(8)Q and equating its scalar and vector parts to
the scalar and vector parts of (8'), we obtain the two relations

- <r,

=/N' - $•,

in which v is the velocity of S' relative to 6" and € our previous

longitudinal stretcher of ratio y = ( i - z>2/^2)"^. Now, since these
relations hold for any value of cr, take first o- = o, and then cr=i,
and remember that, by (9),

<T= -

oV-oo'^y. NI'-NO'= ~v-

Then of the four relations, obtained in this way, one, containing
the n-component of JP -/v, will turn out to be a consequence of
the three others.

These three relations, after a simple rearrangement of terms, and
without Cartesian splitting, give us the required relativistic trans-
formation of the density and the flux of electromagnetic energy and
of the stress in the short form

(10)

The first of these is a scalar equation, the second a vectorial one,
while in the last equation the stress-operator/is written as a dyadic ;
hence the open parentheses. Introducing on both sides any unit

TRANSFORMATION OF STRESS, ETC.

237

:tor n as operand, and closing the parentheses, we obtain the
responding pressure f« =/n, thus :

i/n -/« + j[[3' + u v](vn) + ?($'. <n).

Remember that e is a symmetrical operator, so that (c^'. n) = ($'. en).
To obtain the stress in its more familiar form, take the usual
system of normal unit vectors, i along and j, k at right angles to the
direction of motion. Write in turn n = i, j, k, and remember that
<i = yi, «j = j, «k = k. Then

Splitting each of the stress vectors f15 etc., into its three rectangular
components along the same set of axes, we obtain nine stress
formulae which contract to six, since /12' =/21', etc., and fi^fou etc-
Treating similarly the first two of the equations (10), we have for
the transformation of stress and of flux and density of energy the ten
Cartesian formulae, which were first given by Laue,

/• oy/-/-2t/_. / /-w> »\ r /-/ .- /-/

(ioa)

The transformation formula of g, the electromagnetic momentum per
unit volume, which is simply the energy flux divided by c'\ will be

Applications of the above formulae will be given a little later,
when the domain of their validity has been extended to non-
electromagnetic actions. Meanwhile, notice that the stress, energy

THE THEORY OF RELATIVITY

and momentum, as estimated from the ^"-standpoint, are each built
up of the stress, energy and momentum or energy flux corresponding
to the S'-point of view. This entanglement of the various magni-
tudes, which in classical physics led an independent existence, is
characteristic of the theory of relativity. It is a consequence of
the way in which time and space are involved 'in the fundamental
Lorentz transformation.

In deducing the formulae (10) of transformation of stress and
associated magnitudes, we have used their expressions in terms of
the electromagnetic bivectors, as condensed in (8). Our purpose in
doing so was to show the properties of the simple operator R[ ]L.
But, as a matter of fact, these formulae hold quite independently of
the particular, electromagnetic meaning of f, u and g or Jp/^2. They
are valid in virtue of (3) and (5) alone (with JP = ^2g), that is to say,
for stresses etc. of any origin, electromagnetic or not, provided that
the corresponding ponderomotive force, per unit volume, and its activity
can be represented in the form

p=-v/-i? (A)

(Pp)= -^-^2.divg. (B)

The proof of this statement is most simply obtained by the matrix
method, which in this case is superior to the quaternionic one. Of
course, each method has advantages for certain purposes. In fact,
consider the symmetrical matrix

J\\"> ^12' ./13'

/ 21 ' /22 > f-23 '

f f f

/81» -/32> ./33>

(II)

in which fM=fKt.* Multiply it by, or operate upon it with, the

3333 ,. . .

matrix lor = ^— , =- , ^- , ^ , according to the rule given in the

ox oy oz ol

Note to Chapter V. Then the result will be

* In the case of the electromagnetic field there is a simple connexion between
the matrix (n) and the alternating matrix representing the vectors M, E. See
Note 2 at the end of the chapter.

STRESS-ENERGY MATRIX 239

the constituents of the matrix on the right side being exactly those
given by (A) and (B). This matrix is the equivalent of the physical

quaternion ^=P + -(Pp). We can therefore use for it the same
letter F. Thus, the last equation can be written

(12)

To write this for the force-matrix is exactly the same thing as
to postulate (A) and (B) for the force and its activity.

Let me observe here that the matrix (n)* can be written con-
siderably shorter, thus :

\ 3-

Here one constituent is a linear operator, or, say, a dyadic,
f = i)f1+j)f2 + k)f3, two other constituents are vectors, and the fourth
a scalar. But this heterogeneity of the various constituents of one and
the same matrix need not alarm us. It seems even to harmonize
fully with the original intention of the creation of Cayley, who
wished to see his instrument of multiple algebra treated as broadly
as possible. The only requirement is that the array should be
rectangular. Using the abbreviated form (n#), we have, of course,
to use lor, correspondingly, as the matrix of i x 2 constituents :
V to be applied scalarly, and 3/9 /. In this way we obtain

i & ^S i f • uu\

at once, instead of writing first so many scalar terms and then
gathering them together.

But let us return to our subject. We know already that, whatever
the nature of the ponderomotive force, F is a physical quaternion, or
the matrix F is transformed as | r, / « . And the same thing is true
of lor. Thus, if A be the fundamental transforming matrix, as on
p. 143, we have

F' = FA) lor' = lor A,
and therefore, by (12),

\OT A§' = l

whence, remembering that AA = i,

. (13)

* Which is called Welttensor by Laue and others, but has no particular name
in Minkowski's paper.

240 THE THEORY OF RELATIVITY

Now, substituting here for A the matrix (40), p. 144, remembering
that the transposed matrix A is obtained from A by a mere change
of the sign of ft, and multiplying out the right side, the reader will
easily convince himself of the identity of (13) with the transformation
formulae (io<2), in which jp =<r2g. This proves the above proposition,
which may be restated as follows :

If we make with regard to any ponderomotive forces the
assumptions (A) and (B), or, which is the same thing, the
assumption

then the corresponding pressures, etc., are transformed according
to (loa) or (10), with ^=c-g.

It is, of course, an entirely different question whether those
assumptions are to be considered as universally valid or not.
Assumption (B) is the expression of the principle of conservation of
energy together with the concepts of its localization and flux, and
(A) leads to the principle of conservation of momentum , while there
is a strong tendency among the relativists to retain both of these
principles of classical physics. Thus, M. Abraham uses (12),
involving both principles, in his paper on the electrodynamics of
ponderable bodies,* and appeals to this equation even in his
theory of gravitation, which does not satisfy the principle of
relativity, while Laue makes of it the basis of the general dynamics
of continuous bodies. On the other hand, according to Minkowski's
electrodynamics of moving ponderable bodies, the ponderomotive
force and its activity are expressed by that part of the world-vector
F= - lor § f, which is normal to the four-velocity Y, i.e. by the
matrix

or, which is the same thing, by the physical quaternion

* Rend. Circ. Matem. di Palermo, Vol. XX VII I., 1909, p. I ; ibid., Vol. XXX.,
1910, p. i.

t This <•§> is a non-symmetrical matrix of 4 x 4 constituents, which reduces to
(u) for the particular case of empty space. See Chap. X.

TRANSFORMATION OF STRESS, ETC. 241

and not by F= - lor <S itself. Now, it is true that Abraham's and
Laue's device recommends itself by its simplicity in the case of
the general mechanics of ponderable continua; but, on the other
hand, Minkowski's device seems to offer advantages for a relativistic
theory of gravitation. In fact, a pair of such theories, both satisfying
rigorously the principle of relativity, have been recently proposed
by Nordstrom,* in one of which the four-dimensional force is of
the form lor<S, while in the other it is given by the part of such
a world-vector perpendicular to Y. Now, the latter of these theories
is physically simpler, inasmuch as it leads to a rest-mass inde-
pendent of the gravitation potential, while the former requires the
rest-mass to become an exponential function of this potential.

Certainly, then, the principle of relativity does not compel us to
attribute to the forms (A) and (B) of ponderomotive force and its
activity an universal validity. But it is at any rate interesting to
see the consequences of making the assumptions (A), (B) and of
accepting, therefore, the formulae (10) also for pressures, momentum
and energy of non-electromagnetic nature in any material medium.
Once the reader knows expressly the conditions of their validity,
there is no danger in doing so.

We shall therefore proceed to give here some consequences of the
formulae (10).

Let the system of reference S' be such that there is no flux of
energy with respect to it, i.e. such that $' = 0, and therefore also
y = o. This will, under a restriction, be the case when S' is the
rest-system either of the whole material body, if all its parts have the
same velocity relative to S, or, more generally, of its volume-element
under consideration. We may retain in both cases the symbol v,
which will then generally denote the velocity of an element of the
body with respect to S. The restriction hinted at consists obviously
in supposing that in u are contained only such kinds of energy as do
not flow through the element in question, e.g. energy of elastic
deformation, energy stored up in the atoms, heat for the case
of uniform temperature, and — as Laue adds — 'possibly also some
new kinds of energy, yet undiscovered.' But to these the electro-
magnetic energy cannot generally be added, since it may flow even
in the rest-system. If, however, we have in S't say, an electrostatic

* G. Nordstrom, * Relativitatsprinzip und Gravitation,' Phys. Zeitschrift, XIII.,
1912, p. 1126. See also M. Behacker, ' Der freie Fall und die Planetenbewegung
in Nordstroms Gravitationstheorie,' ibidem, XIV., 1913, p. 989.
S.R. Q

242

THE THEORY OF RELATIVITY

or a magnetostatic field, we can include in it the density of the
corresponding energy, combining at the same time the electric
or the magnetic stress with the mechanical one.

Keeping this in mind, and writing <r2g for J3, we obtain, for the
density of energy and of momentum and for the stress, as estimated
from the ,S-point of view, the formulae (10), considerably simplified,

(14)

In Cartesians, with axes taken along the velocity v of a particle and
at right angles to it, these formulae are, as (ioa) without the dashed
fluxes of energy,

=-/23 > JZ\ ~ 7/31 ') fl2 ~ 7/12

We may notice in passing that the sum of the diagonal constituents
of the matrix <S, i.e.

/11+/22+/SS-** (T5)

is always an invariant. With the above choice of axes, we have
also separately, by (140) or by (io«J,

The invariant (15) vanishes in the case of purely electromagnetic
Maxwellian stress. But for mechanical stresses its value will in
general differ from zero.

In order to understand the physical applications of the formulae
(14), we require still a certain explanation. The stress denoted in
these formulae by f, and called the 'absolute' stress, must be
carefully distinguished from the elastic stress as usually employed,
which, in the writings of Abraham, Laue and other authors, is given

RELATIVE STRESS 243

the name of relative stress. According to Laue, we have to put, for
every dynamically complete system, F= o, and to write, therefore, at
the head of dynamics of continuous bodies, the equation*

o. (16)

This amounts to putting P = o in (A) and (B), so that

At the same time it is assumed that the resultant force acting upon
any individual portion of the body in question is given by

: . N=f' <•»>

where G= \gdS, the integral being taken throughout the volume of

that portion. If, therefore, dS is an individual volume-element
of the body, the relative stress which we shall denote by /, the
symbol of an operator,! will, according to the familiar definition,
be given by

-

or

, (a)

where P^/i, etc., and where — is the individual rate of change.

On the other hand, the meaning of the absolute stress /is given by
the second of (i6a) or, in expanded form, by

dg afx 3f2
-~~"

where ^- is the local rate of variation, corresponding to constant

* Laue's symbol equivalent to lor in this connexion is Az'u, a four-dimensional
' scalar divergence,' identical with Sommerfeld's Div.

t So that /n = pn will be the pressure, per unit area, upon a surface element
whose normal is n, and pn\, pnii Pns the rectangular components of this pressure.
As will be seen presently, p is, unlike f, a non-symmetrical operator.

244 THE THEORY OF RELATIVITY

values of x, y, z. Now, we have, for any orientation of the system
of rectangular axes,

and therefore, comparing (a) with (<£),

Vl = fl-gvl, p2 = f2-gz>2, p3 = f3-gz>3, (i8a)

i.e. for any direction of n,

pw = fn-g(vn).

Omitting the operand n, we may write this result, in terms of the
stress-operators themselves,

/=/-g(v . (18)

This is the required connexion between the relative stress p and the
absolute stress / Notice that, / being symmetrical or self-conjugate,
p is in general non-symmetrical, since g may differ in direction
from v. Thus, for instance, Pyt=f\i- gflv while Ai=/i2~<^i^2-
Only when g || v does the relative stress become self-conjugate.

Let us now return to (14). Remember that, for the rest-system,
P'—f't write down g(v by the second of those formulae, and
subtract it from the third one. Then the terms containing u will
cancel one another, and the result will be

or, if i be the unit of v,

/ = e/€-/32y.€/i(i . (19)

Such then is the transformation formula of the relative stress. The
reader will find no difficulty in splitting (19) into nine Cartesian
equations for /n, /12, etc., especially as this procedure has been
illustrated a moment ago by the passage from (14) to (140). It is
interesting to remark that/ depends only upon /' and the motion of
the element in question, but not upon u', the density of energy.
And, whenever p' = o, we have also / = o. This, besides the original
definition (#), is the reason why the relative stress p (and not the
absolute one) is considered as the stress.

The simplest case occurs when the body, viewed from the rest-
system, is subjected to what is called a hydrostatic or isotropic

ENERGY AND MOMENTUM 245

pressure (i.e. to a pressure which is purely normal and equal for
all directions of n), either uniform or varying from point to point.
Then the stress-operator /' degenerates into an ordinary scalar, the
pressure in the more familiar sense of the word.* In this case /'
can be written before the stretching operator, so that (19) gives
at once

Now, e2 = y2i(i+j(j +k(k , and y2 -{Py2 = i, so that the right side of
the last formula is, in Gibbs' terminology, an idemfactor, i(i + j(j + k(k ,
leaving unchanged any operand n whatever. The result, therefore,
is that

#=/,

or that isotropic pressure is a relativistic invariant. This result was
first obtained by Planckf from thermodynamical considerations aided
by the principle of relativity, then by Sommerfeld + from what he
believed to be a purely geometric enunciation of the behaviour of
four-dimensional vectors and their projection, and, finally, by Laue,
whose method has been here adopted. The reader will find it worth
his while to compare the latter with the two former methods, and
is for that purpose referred to the papers of Planck and Sommerfeld
just quoted.

So much as regards the stress and its transformation. Next,
consider u and g, the densities of energy and of momentum for
which the first pair of (14) hold. In these formulae we have only
to substitute the identity /' =/'. Thus, taking i along the direction
of motion of the given element of the body, we have in general,
that is to say, for any elastic stress /',

* = 72K + £2Ai'] (2°)

and

(21)

where /'v is the same thing as zip/, of course.

Let dS be the rest-volume of an element of the body, and con-
sequently dS=dS'jy its ^-volume. Then we shall have for the

* Reckoned positive if pressure proper, and negative if tension proper, as
before.

t M. Planck, < Zur Dynamik bewegter Systeme,' Ann. der Physik, Vol. XXVI.,
1908, pp. 1-34.

%Ann. der Physik, Vol. XXXII., 1910, p. 775.

246 THE THEORY OF RELATIVITY

energy of that individual element, as estimated from the S-point
of view,

To obtain the whole energy U, this is to be integrated throughout
the body. Generally speaking, there will be no simple relation
between U and U' . For, even if z/,and the stress were constant
throughout the body, the value of /3 and also the direction of v
may change from point to point. And if but one particle of the
body moves with varying velocity, then the velocity will also, as a
rule, vary from particle to particle. Let us suppose, however, that
this heterogeneity of the inner state (#', p') and of the motion of
the body can be neglected. Then, if V and V be the volumes
of the whole body from the two standpoints, its total energy, as
estimated by the ^S-observers, will be

<7=7(<7' + /^/nF'). (200)

We shall return to this formula presently, in order to compare the
difference U- U' with the expression of kinetic energy given, for
the simplest particular case, in Chapter VII.

Treating similarly the equation (21), and making the same assump-
tion of homogeneity, or considering the whole body as a particle,
we have, for its total momentum,

We have seen in Chap. VII., formula (24), that, according to
Minkowski's dynamics of a particle, the momentum of the particle
would be simply ym times its velocity, where m, the rest-mass of
the particle, is an ordinary scalar magnitude. Thus, according to
that manner of treatment, the momentum would always coincide in
direction with the velocity. This isotropic behaviour of the rest-mass
appears now as the simplest particular case of formula (21 a), which
holds for a particle conceived as the limit of an extended body.
We can still write

but now m, instead of being a simple scalar, will be a linear vector
operator, e.g.

*—2+"fff* (22)

so that the momentum will generally differ in direction from the
velocity.

INERTIA OF ENERGY 247

The first part of m is an ordinary scalar, namely

This is the expression of the famous inertia of energy which, as a
consequence of the principle of relativity, has been enunciated by
Einstein.* If a body gains or loses n ergs of energy, say, in the
form of heat, then we have to look for an increase or diminution
of its rest-mass by - io~20 grams. The second part of m is due to

the stress. Since /' is, in general, an operator, this part of m will
also be an operator. It will be remembered that/', being identical
with the original /', is self-conjugate. The stress, therefore, will
have three mutually perpendicular principal axes. Let these be
represented by the unit vectors a, b, c, each of which can be taken,
of course, in both its positive and negative sense. And let us
denote the corresponding principal pressures, which are ordinary
scalars, by /„', /ft', pcf. Then, if v is along a, for instance, we shall
have

since ea = ya. Similarly, if the body happens to move along b or c.
Thus, the principal axes of the mass-operator m coincide with the
principal axes of the stress. \ The corresponding principal values
of the rest-mass are

(»*«)

*Cf. Einstein's papers in Ann. der Physik, Vol. XVIII., 1905, p. 639,
Vol. XX., 1906, p. 627, but especially ' Ueber die vom Relativitatsprinzip gefor-
derte Tragheit der Energie,' ibid.y Vol. XXIII., 1907, p. 371. Independently
of the principle of relativity, the inertia of energy, in the case of radiation, appears
in a valuable paper of K. v. Mosengeil, Ann. der Physik, Vol. XXII., 1907,
p. 867. The history of this concept can, of course, be traced a long way farther
back. Its origin can be looked for in Maxwell's pressure of light, and in con-
nexion with this many English physicists spoke about 'momentum carried by
light waves ' a long time before the theory of relativity arose.

f This coincides with Herglotz's result obtained by a different method : Ann.
der Physik^ Vol. XXXVI., 1911, p. 493. The reader will find in this beautiful
paper a systematic development of relativistic mechanics of deformable bodies.

248 THE THEORY OF RELATIVITY

The momentum is parallel to the velocity of the body when and
only when it happens to move along one of its principal stress-axes.
Notice that, by what has been said, this anisotropy would be a
property of the rest-mass itself. When, therefore, we pass to consider
the acceleration of such a body, or particle, in relation to the moving
force, according to the equation of motion

Jly^v^N, (23)

we can no longer express the inertial behaviour of the body in terms
of a ' longitudinal ' and a ' transversal ' mass, as in Chapter VII. The
axial symmetry produced round v in that comparatively simple case
was due to the assumption of a scalar rest-mass. The case now
before us is much more complicated. Even if the inner state of
the body is supposed to remain invariable, a full description of
acceleration in connexion with force requires a linear vector operator,
involving six scalar inertial coefficients. The dynamics of trans-
lational motion of such a body is, obviously, entangled with the
dynamics of its rotations. Unlike classical mechanics, these two
kinds of motion cannot, rigorously speaking, be treated separately.
It can be shown, by considering the moment of momentum, that to
maintain such a body in uniform rectilinear motion, a certain couple
is required. Only when the constant vector-velocity v of the body
coincides in direction with one of its principal stress-axes, would the
moment of this couple vanish. Again, suppose that there is no
impressed resultant force, i.e. that N = o. Then the momentum will
be constant in both size and direction relative to S, say, equal C, and

If, therefore, the body rotates together with its stress-axes, the
motion of translation will not be uniform and even not rectilinear.
Notwithstanding the absence of a resultant *S-force the body may
move with varying velocity relative to the framework ,5. And it
will do so if, for instance, its initial velocity does not coincide in
direction with one of the principal stress-axes and if the couple
mentioned above is not applied. But we cannot dwell any longer
upon this curious subject.

All that has just been said with regard to the anisotropy of
rest-mass has, at least for the time being, merely a theoretical
interest. In fact, nobody has ever observed in translational inertia

ANISOTROPY OF INERTIA 249

any departure from isotropy. On the other hand, it must be
confessed that no phenomena of this kind have been sought for
expressly and that direct comparisons of inert masses (i.e. apart
from gravity) could not easily be made more accurate than to one
in ten or hundred thousand parts. One thing, at any rate, seems
certain : If the above formulae are accepted, we cannot reasonably
hope to produce observable anisotropy of mass by artificial pressures
or tensions in any lump of matter. For, according to (220), hundreds
of atmospheres appropriately applied would produce a departure
from isotropy of mass amounting only to io2. ioV~2==io~13 of a
gram per cubic centimetre. But for all that we know there might
be anisotropy of inertia in natural crystals, corresponding to some
enormous 'latent stresses.' And to embody such stresses into /'
seems no less, and no more, legitimate than to condense in U" so
much * latent energy ' as is necessary to account for the observable
mass of a body. But, apart from any theory, experiments on
crystals seem worth trying, whether to reveal some traces of
anisotropic inertia or to push it below a numerically definite
limit.*

Of course, if it is assumed that the stresses represented by /' are,
under all circumstances, only of the order of manifest tensions and
pressures known as such from experience, then the influence of
the differences /0' -/b', etc., upon inertia will be far too small to be
ever detected. But if so, then there will be also no sensible
contribution of stress to inertia at all. Such, in fact, is the prevailing
opinion.

According to this opinion the stress-term in (22), (21) and, for
slow motion, a fortiori in. (20), where it appears with the coefficient /32,
can be omitted for all ordinary material bodies. But the case is

*In connexion with this subject, Prof. A. W. Porter of University College,
London, draws my attention to experiments made by Poynting and Gray,
who tested for anisotropy of gravitation between two quartz spheres (Phil.
Trans., 192, 1899, A. p. 245 ; cf. also Poynting and Thomson's Text- Book of
Physics, Properties of Matter, London, 1909, p. 48). Their results showed that
this anisotropy could not amount in one case to more than one part in 2800, and
in another case to more than one part in 16000. On the other hand, proportionality
between mass and gravitation, first tested by Newton in his pendulum experiments
and carried to further refinement by Bessel, has been more recently shown by
Eotvos (Math, und naturwiss. Berichte aus Ungarn, Vol. VIII., 1890) to be
true to one part in ten millions, in the case of isotropic bodies at least. So far as
we know, experiments of this kind have not yet been made with crystalline bodies,
but are now under consideration at University College.

250 THE THEORY OF RELATIVITY

different, of course, when the energy and the stress are purely
electromagnetic, when the ' body ' becomes simply a region of space
containing an electromagnetic field. Under these circumstances the
part played by p' is no longer negligible, unless we wish to neglect
the whole mass m, and therefore also the whole momentum. In fact,
not only then are the pressures or tensions /a', etc., of the same
order as the density u' of electromagnetic energy, but some of them
can even wholly annul the contribution of energy to mass. Let,
for instance, the field in S' be a homogeneous electrostatic field
E' = const., such as is contained between the plates (discs) of a plane
vacuum-condenser, far enough from the edges of the plates. Then
u' = l>£'2, and if a be taken along the axis of the condenser or along
the Faraday tubes, /a', being a tension proper, is equal to - ^£'2,
while /6', //, being pressures proper, are each equal to J^"2. There-
fore, by (2 2 a),

while

Thus the condenser, apart from the plates, has equal rest-masses in
all transversal directions, while its longitudinal principal rest-mass
vanishes altogether. If it is moved along the tubes it has no
momentum. This property, which holds separately for each length-
element of a Faraday tube, harmonizes with Sir J. J. Thomson's
well-known representation. The tubes may be straight, as in the
above case, or curved and of varying section. The only condition
being that there shall be no flux of energy in S't we can certainly
apply the above reasoning to any electrostatic field. Summing up
the contributions due to the elements of infinitesimal filaments (with
appropriate consideration of their directions), the mass-operator of
the whole field can be found. If the field is radial and symmetrical
round a point O\ as in the case of the Lorentz electron, the mass-
operator m degenerates into an ordinary scalar, the rest-mass of the
electron, or rather of its whole field. The reader is recommended to
prove this in detail, and to compare the result to be thus obtained
with the formula of the electromagnetic rest-mass given previously.*

* The above dynamical considerations have also an important bearing upon the
theory of the celebrated condenser-experiment of Trouton and Noble (Proceedings
Roy. Sac., Vol. LXXII., 1903), in which a second-order moment of rotation on a

SCALAR REST-MASS 251

Let us now once more return to stresses and energies of any
origin. In the simplest case of hydrostatic or isotropic pressure^
whatever its order of magnitude, our above /' degenerates into an
ordinary scalar, so that, in (210), 7~1e/'v = y~1ev./' = v./', while, in
(2o<2),/n'=/', and therefore

(24)

These are Planck's formulae (loc. cit.}. Since isotropic pressure is an
invariant and V— f'/y, we have also

x=u+pv=y(U'+p'v') = rx;t (25)

where x'> tne rest-value of x> is Gibbs' 'heat function for constant
pressure 'or enthalpy.* The momentum is now in the direction of
motion. The mass-operator (22) degenerates into

the scalar rest-mass.

Thus, in the case of isotropic stress, the inertial behaviour of the
body, or particle, is characterized by a simple scalar, as in Chap. VII.
But still the rest-mass will in general vary in time, inasmuch as the
inner state of the particle (£7", /', V) may undergo changes during
its motion. If this is the case, e.g. if the enthalpy of the particle
varies, then SXYC does not vanish, or, in other words, the Min-
kowskian four-force X is no longer perpendicular to the particle's
world-line. In fact, instead of equation (20), p. 194, we now have

dY ,dm

suspended condenser due to the earth's orbital motion was sought for. But
a somewhat thorough exposition of this subject would be beyond the limits and
purposes of the present volume, and the interested reader must therefore be
referred to § 18 of Laue's book already quoted. Here it will be enough to say
that the relativistic theory accounts fully for the negative result of the Trouton-Noble
experiment.

* The latter name is used by the Dutch school of physical chemists, while the
name nearly always used in England is total heat.

252 THE THEORY OF RELATIVITY

and consequently, since /' can be written for the proper time T,

SXY= YY*— = -Pd—
or, by (26),

This proves the statement. Developing the left-hand side, by (18),
(17^), p. 193, we have, in terms of the Newtonian force N and the
velocity v of the particle,

(Nv) = |(^y)-^> (27)

or also, by (25) and (26),

This is now, instead of (22), p. 194, the equation of energy.

To see its meaning, consider the particular case of constant
pressure, or what may be called isopiestic motion. Then, if H be
the heat communicated to the particle per unit /'-time,

the heat supply being estimated from the point of view of the
system S' in which the particle is instantaneously at rest. Con-
sequently,

/_, v i ,, dU dV i ox

<Nv>+^=^+^-

The first term on the right is the rate of increase of the total energy
of the particle, the second term gives the work done per unit time
by the particle in expanding, while (Nv) is the activity of the
impressed force, everything being estimated from the ,$point of view.
If, therefore, (28) is to express the conservation of energy in S, just
as (28') does with respect to S', we have to write for h, the rate of
heat supply as estimated from the .S-point of view,f

*This result can be verified at once by multiplying eq. (23) of the present
chapter scalarly by v.

1 It is scarcely necessary to warn the reader that h is not equal to dx\dt. It
becomes so (for constant pressure) only in the rest-system. Putting in (28) v=o,
7=1, we obtain (28').

HEAT SUPPLY 253

And, since dt=ydf, we have to require that the relativistic con-
nexion between corresponding infinitesimal amounts of heat supplied
or withdrawn shall be

SH=-SJf'. (30)

This transformation formula agrees entirely with what follows from
Planck's thermodynamical investigation. In fact,* one of Professor
Planck's most fundamental results is that entropy is invariant with
respect to the Lorentz transformation,

and another of his results states that temperature is transformed like
volume,

0^-ff.

7

Now, the temperature being here defined in the well-known thermo-
dynamical way, we have, for reversible heat supply, &H' = 0'di],
and on the other hand (granting that a process reversible in
S' is also reversible from the ^"-standpoint), 8If= 6 dt], whence

But, instead of recurring to temperature and the second law of
thermodynamics, the transformation formulae (29) and (30) can
equally well be considered as consequences of the principle of con-
servation of energy combined with (28), which in its turn is a
consequence of the equation of motion (23) and of the relativistic
behaviour of momentum. Whatever the logical order of exposition,
the important thing to notice is that the several properties are con-
sistent with one another.

Before leaving the discussion of variable rest-mass, only one
more remark. It has been shown in Chap. VIII. that the electro-
magnetic ponderomotive force per unit volume plus i/c times its
activity is a physical quaternion. In agreement with this the total
force N of Chap. VII. had the property that y[i(Nv)/<r+N] was a
physical quaternion. Both of these were particular instances of a
more general property which is now before us. When the moving
body receives or gives up heat, or more generally, when its enthalpy

* Cf. Planck's paper quoted on p. 245. Unfortunately, there is in this book no
place for an adequate discussion of the foundations of relativistic thermodynamics.

254 THE THEORY OF RELATIVITY

is varying, then the above expression is no longer a physical
quaternion. What now continues to be such a quaternion is

or

In Chap. VII. we had simply — (mc^y) — (Nv), while now we have,

instead of this, the equation (27). Thus, in general, for any motion
of the body,

is a physical quaternion, and more especially, for isopiestic motion,

?. (31)

In all such cases, therefore, we have to add to the activity of the
impressed force the amount of heat supplied to the body per unit
time. This property will reappear, in the next chapter, in connexion
with Joule's heat in electrical conductors.

If the enthalpy x', and therefore also the rest-mass of the body, is
kept constant, we fall back to the simple case treated in Chapter VII.
The activity then becomes, by (27),

identical with (22), p. 194. Using the form (27^), we may also
write, equivalently,

,__ d dU dV

which reads : Work done upon the body = increase of its energy plus
work done by the body in expanding. The corresponding condition

X' = U' +p' V = const.

can still be fulfilled in a variety of ways. Thus the motion may be
adiabatic as well as isopiestic. Or, we may give up both of these
conditions and suppose instead that Vdp ' jdt' is just balanced by
the (positive or negative) heat supply ti. Or finally, the inner state

*

CONSTANT REST-MASS 255

of the moving body may be invariable, i.e. U', p' as well as V may
be kept constant. But even then the work done by expansion does
not disappear from (320) unless the motion is uniform. For, with
constant V, we have

dV dy-i

dt~ dt '

which expresses the varying FitzGerald-Lorentz contraction. But
whatever the way in which \' is kept constant, we have the same
equation of motion as in Chap. VII.,

and consequently the longitudinal and the transversal masses return
to their rights, being again given by

ml = my3, mt = my,
where m has now the explicit meaning

X' U'+p'V
»~$~—-g— (33)

Finally, if the pressure /, and therefore also p, is assumed to vanish,
the equation of energy becomes

and the constancy of the rest-mass

U'

m = 7^

means constancy of the particle's store of energy. In this case the
difference between the energies U and U' can be looked at as
entirely due to the motion of the particle and called its kinetic
energy relative to S. The value of the kinetic energy thus defined
is identical with that given on p. 195. In fact, the first of (24)
becomes now U=yU', so that

But, speaking rigorously, ' kinetic energy ' is now deprived of
the distinct part it played in classical mechanics. And its entangle-
ment with other kinds of energy becomes even more intricate when
we pass from this simplest case to any of the preceding ones.

256 THE THEORY OF RELATIVITY

It may be useful to illustrate here the mass formula (33) by a few
numerical examples. Thus, taking 2-1 gram calories for what is
called the solar constant (energy received from the sun per minute
per cm2, at the earth's mean distance), we have for the sun's total
radiation per minute

477(1-5 . io13)2. 2-1 . 4-2 . io7 ergs,

so that the diminution of the sun's mass due to radiation would be,
per minute, 2-8 . io14 grams, and per year

Sm= 1-5 . io20 grams.

This seems at first a prodigious loss ; but the sun's mass being
2 . io33 gr., the proportionate loss per year,

is quite insignificant. Next, take the example adduced by Planck.
A mixture of 2 gr. of hydrogen and 16 gr. of oxygen develops in
the act of producing water, at ordinary pressure and temperature,
2-9 . io12 ergs of heat; the corresponding diminution of mass amounts
to 3-2 . io~9 gr., and the proportionate loss due to this intense
reaction,

8m

— =2 . I0~10,

m

would again be far too small to be observed. Numbers of similar
order would result for other instances of chemical reaction.
In short, the ' latent energy ' which (if we neglect the contribution
due to stress) is to account for mass does not manifest itself in
any one of those processes in which atoms are implied as wholes.
We are thus driven back to the interior of the old chemist's
atom, and have to look for that energy in the disintegration of
atoms known in connexion with radioactive phenomena. In fact,
if we are to judge from their observed heat-effect only, the
amounts of energy developed in such processes exceed immensely
all those liberated in ordinary chemical reactions, and Professor
Planck seems to see in radioactivity a kind of verification of
the energetic theory of inertia. Now, it is true that these pro-
cesses have disclosed to the physicist atomic stores of energy of a
copiousness not even suspected a short time ago. But, not-
withstanding their unparalleled vigour, radioactive phenomena reveal
but a very minute fraction of the assumed latent energy. Thus, to

LATENT ENERGY 257

quote Planck's own example, one gram-atom of radium would lose
through its heat production (30240 gram calories per hour) -012
miligr. of its mass per year; the proportionate loss, therefore,
amounting to

8m/m = -5 . io~7 per annum,

is again too small to be observed.

We may add that the latter 8m is small even when compared
with the mass disintegrated during the same interval of time. For
this amounts, per 225 gr. of radium present, and per year, to
= 9- io~2 gr., so that, in round figures,

The mass of the disintegrated parent substance reappears sensibly
undiminished in the masses of the descendants.

Thus, even radioactive phenomena reveal to us practically
nothing of the assumed latent energy c-m. Its bulk remains as
latent as anything ever was. It must, therefore, be confessed that
the energetic theory of rest-mass, attractive and promising as it may
seem, has for the time being the character of a purely formal reduc-
tion of one concept to another. Nobody doubts, of course, that the
chemical atoms are themselves exceedingly complicated systems, and
that there are therefore many ways left of throwing the chief stores
of latent energy upon a host of ultra-atomic entities, electrons or
what not. If so, then some spontaneous disintegration, affecting the
atomic structure even more profoundly than that which in our days
is associated with the name of radioactivity, may induce the gates of
those copious stores to open to the human eye. But as yet we have
not the least knowledge of such phenomena. It is for this reason
we have said that it is equally legitimate to assume latent stresses
along with the manifest ones in the mass formulae as to assume
latent energies. Both are originally denned only by their variations,
in time and in space respectively. And, for the present, both
would have a purely formal character.

The above mechanical, and partly thermodynamical, subjects
have been treated at some length because of their affinity with
the fundamental electromagnetic equations for vacuum. Returning
now once more to Electromagnetism, we shall close this volume
by dedicating the next chapter to Minkowski's equations for
ponderable media. _

S.R. R

258 THE THEORY OF RELATIVITY

NOTES TO CHAPTER IX.

Note 1 (to page 234). Let i, j, k be the antecedents, and fl5 f2, f3 the
consequents* of the stress-dyadic/. Thus, if/ as in equation (5), is to be
applied as a post-factor,

/=i)f1+j)f2 + k)f3. (a}

This means that i/r=(ii)f1 = f1) etc., and in general,

which is equal to fn, as it should be. Similarly, writing instead of n the
Hamiltonian V,

V/=(Vi)f]+(Vj)f2+(Vk)f3

= ^1 + ^ + ^3. Q.KD.

d.r 9j ?)z

Using the notation of Gibbs, Scientific Papers, Vol. II. p. 76, we should
write /= i^ + jf2 + kf3 , and

c)f 9f „ ,

where the dot means scalar application of V. But since, in our case, the
prescription of applying V scalarly is already given by the open parentheses
in the dyadic (a\ we do not require the dot or any other symbol of scalar
multiplication.

Note 2 (to page 238). Let h, as in Note 6 to Chapter VIII., be Min-
kowski's alternating matrix equivalent to the electromagnetic bivector,
i.e. let, according to (c\ p. 230,

o,

(«>

Multiply it into itself. Then the first constituent of the first row of the
resulting matrix hh will be

where fn is the corresponding component of the Maxwellian stress and
X=%(M2-E2) the electromagnetic Lagrangian function per unit volume.
Similarly,

= -/22 - A, etc., (M)^ =tt-X,

* This is Gibbs' nomenclature.

STRESS-ENERGY MATRIX

259

where u is the density of electromagnetic energy and g that of momentum,
as throughout the chapter. Thus

-(hh\.

A, /12» /13>

21» /22+A, /23»

31» /32> ./33+A,

where <S is the matrix defined by (11), p. 238, and A is written for
A times the unit matrix of 4 x 4 constituents. The required connexion is,
therefore,

cS=-^-A. (b)

It will be remembered that A. is one of the invariants of h. And, since
h' = Ah A, the last equation gives at once

in agreement with (13), p. 239. On the quaternionic scheme we have,
instead of (b\ the operator R[ ]L of an analogous and somewhat simpler

structure.

CHAPTER X.

MINKOWSKIAN ELECTROMAGNETIC EQUATIONS FOR
PONDERABLE MEDIA.

IN Minkowski's notes, which after his death were worked out by Dr.
M. Born,* the electromagnetic equations for moving bodies, satisfying
rigorously the principle of relativity, are deduced in a very ingenious
way from the fundamental equations of the electron theory. And
since the electronic equations were previously known to be invariant
with respect to the Lorentz transformation, and gave to the relativist
his first standard magnitudes, such a deduction was certainly very
desirable and interesting. In fact, it occupied Minkowski's thought
vividly during his last days. But in his own paper of 1907, re-
peatedly quoted, Minkowski adopts a purely phenomenological
method, and deduces the equations for moving bodies, now gene-
rally associated with his name, from Maxwell's equations for stationary
media by subjecting them to a Lorentz transformation.

In the present chapter we shall avail ourselves of the latter method
only, which, apart from other considerations, recommends itself by
its mathematical simplicity. Readers, and especially those who
desire to see the electron theory made the foundation of all electro-
magnetic science, are referred to Dr. Bern's paper just quoted, where
the resulting equations! are wholly identical with Minkowski's
original equations to be given presently.

* Fortschritte der math. Wiss. in Monographien , edited by O. Blumenthal,
Heft I, Teubner, 1910, p. 58.

t E.g. the differential equations of the field and the connexions between the
various vectors involving the specific ' constants ' of the medium, but not the
formulae concerning ponderomotive action. These, as far as I know, have not
yet been worked out electronically, for moving bodies. Einstein and Laub give
an electronic deduction of the ponderomotive forces upon stationary media in

PONDERABLE MEDIA 261

We shall retain here the notation adopted in Chapter II., where
Maxwell's equations for a perfect insulator are collected under (3),
p. 26. In the more general case of a conducting body, we have to
supplement the displacement current by the conduction current.
The latter, reckoned per unit area, we shall denote by I', and the
electrical conductivity by <r. Thus, Maxwell's equations, written for
the system S\ in which the ponderable body is at rest, will consist of
the two groups :

+ 1' = c. curl' M' ; div <T
= -r.curl'E'; div'JE'

independent of the properties of the particular body, and

JE' = /*M', I' = o-E', (2')

containing its specific 'constants.' These, the permittivity, inducti-
vity (or permeability) and conductivity, which hereafter will play the
part of invariants,* may be either simple scalars (more generally,
linear vector operators) if dispersion is disregarded, or otherwise
compound differential operators. In the latter case (in which practi-
cally K alone is concerned) the operator K is to be expressly
constructed so as to be invariant. Thus it may consist of derivations
of any order with respect to the proper time of the body.

In what follows we shall limit ourselves to isotropic media, so that
K, n, cr will have at any rate a scalar character, being either scalar
magnitudes or scalar operators involving differentiations.

Let now 6" be another system of reference (say, the earth), relative
to which our ponderable medium, together with its rest-system S',
moves with a uniform velocity v. Assuming the rigorous validity of
Maxwell's equations (i') and (2') in S', and subjecting them to the
appropriate Lorentz transformation, we shall obtain two groups of
equations for the ^-standpoint. Call them (i) and (2). What
properties are we to require from (i) and (2) in the name of the

Ann. der Physik., Vol. XXVI. 1908, p. 541. Their formulae coincide, in the case
of non-magnetic bodies, with those given by Lorentz in his article in Encykl. der
math. Wis sense haft en, Vol. Vg. 1904, pp. 245-250.

* This means that if the body is at rest in any other legitimate system S"t the
connexions <Ew=A'B*f etc., hold again with the same A", /*, <r.

262 THE THEORY OF RELATIVITY

principle of relativity? In the previous case of vacuum, when
there was nothing to be carried along with the observers, all legi-
timate systems, S, S\ S", . . . were wholly equivalent to one another,
and the relativistic requirement was simply invariance or preservation
of form of the equations. The case before us is different. The
ponderable dielectric, with its specific properties, is at rest in one
system at a time, and moves relatively to all other systems. The
rest-system, in our concrete case S', is an uniquely privileged frame-
work. If, in other concrete cases, the body were fixed in S or in S",
and so on, we should have to require the non-dashed or the double-
dashed equations to be of the same form as the above (i') and (2').
But, S being a system, relative to which the body does move
(uniformly), we have to require only that the groups of equations
(i) and (2), which might both contain the velocity v,* should be
invariant with respect to the Lorentz transformation by means of
which we pass from S to any other legitimate system. If this require-
ment were not fulfilled, Maxwell's equations could not be used for
relativistic purposes at all. But, as a matter of fact, they stand this
test completely.

It seemed advisable to dwell a little upon these explanations ;
firstly, to avoid possible misunderstanding, and secondly, because the
procedure and the test here exemplified are of general importance.
They are the same in every other case in which the relativistic
equations to be constructed concern any phenomena in ponderable
bodies.

In order to obtain the two groups of equations, numbered in anti-
cipation (i) and (2), and to see at the same time their invariance, put

and

and similarly for the non-dashed letters. Further, introduce the
quaternion

and write, as throughout the book, D' = 3/3/' + V, using this operator
as both a prefactor and a postfactor, as explained on p. 223.

* Though, as a matter of fact, only one of them will do so.

PONDERABLE MEDIA 263

Then the four equations (i') will assume the quaternionic form

These are identical with the first and the second pairs of (i')
respectively.

Now, let Q[ ]Q be our usual transformer from 6" to S', and there-
fore Qc[ ]<2c the inverse transformer. Apply the latter to each of
the equations (I'a) and insert QcQ=i between D' and f,' (or L'),
and QCQ= i between JT (or E') and D', in very much the same way
as on p. 208. Then the result will be

and similarly for the second equation, i.e.

2Cn

(la)

where $=<2$'<?c> yi=QM'Q> an<* similarly for the other pair of
bi vectors, and C=QCC'QC. Conversely,

S'=aSC, etc., C=QCQ.

In short, C=ip + I/c is a physical quaternion, JJ and L are left-
handed physical bivectors, and Ji and R right-handed ones.* C may
be called the (macroscopic) current-quaternion, while the electro-
magnetic bivectors need no special names.

Now, the ^-equations (la) are precisely of the same form as those,
(i'tf), for the rest-system. And so they will be also for every other
legitimate system of reference. The velocity of the body does not,
in fact, enter into these differential equations at all. We can now
pass from their quaternionic form (la) to the vectorial one, and shall
thus obtain the required first group of equations :

-^- + ! = <:. curl M, etc., (i)

exactly as in (i') without the dashes. At the same time we have
proved their invariance with respect to the Lorentz transformation.

* It will be remembered that the latter property is a necessary consequence of the
former. In fact, as was proved in Note 5 to Chap. VIII., p. 229, if A-iB is a
left-handed, then A + iB is always a right-handed bivector.

264 THE THEORY OF RELATIVITY

This property finds its immediate expression in the above quater-
nionic form (i<z).*

Moreover, the stated transformational properties of the electro-
magnetic bivectors and of the current-quaternion lead at once to the
second group of equations for the moving body, to be deduced from
the Maxwellian connexions (2'). In fact, since both L = Jtl- tEand
|JJ = M-i(£ are left-handed bivectors,! we have in exactly the same
way as on p. 210, writing again e for the longitudinal stretcher of
ratio 7 = (i - #2/r2)~^,

whence, by the first and second of the connexions (2'), and after
an easy rearrangement of terms,

Jtt~

Both of these relations, involving the substantial properties of the
medium, contain its velocity. Again, since C=tp + I/c is a physical
quaternion, we have, by (i'^) of Chap. V., p. 125,

I = el' + y/o'v
and

whence, by the last of (2'),

* Or in Minkowski's matrix form. This consists of the two equations
\or/i=-s, \orH*-0,

in which s is the matrix-equivalent of C, h and H the alternating matrices corre-
sponding to the bivectors |C and L respectively, and H* the dual of H.

t Notice, in passing, that this being the case, |£2 and L2 are complex in-
variants. These split into the four real invariants,

PONDERABLE MEDIA 265

But -E' = -E+ -VvJE. Hence, after a slight rearrangement of
terms,

Thus I appears as the sum of the convection current /ov and the
conduction current, for which we have written 3E, the latter being
proportional to the conductivity.*
Using the convenient abbreviations

Ex = E + -VvJE, <£x = <£ + - VvM

c c , ^

(A)

and gathering together the above results, we obtain the required
second group of equations, valid from the standpoint of the
system S,

I-/>V = E = e^E*. I

These three connexions involve the velocity of the ponderable
medium relative to that system. It remains only to prove that
they are invariant with respect to the Lorentz transformation.
Now, introducing the velocity-quaternion

we have, identically,

* Adding the displacement current, we should have

the 'total' current. This is, by the first of (i), always solenoidal.

266 THE THEORY OF RELATIVITY

and each of these expressions * is a physical quaternion, ^L q.
Moreover, starting from the current-quaternion C and its conjugate
Cc, we easily obtain the identical equation

'-[C+ 1 YCC Y] = e2! - pf-v + i[(Iv) -

of which the left-hand side is, obviously, again a physical quaternion.
So also is its right-hand side, which, by the third of (2), is equal
to a-rj. Using, therefore, the above identities we can write the
whole of (2), in terms of physical quaternions alone,

(20)

This proves the in variance of the relations (2) with respect to
the Lorentz transformation.! Thus the whole of equations (i)
and (2) satisfy the principle of relativity. Q.E.D.

It is worth noticing here that the world-vector corresponding
to the quaternion

is the part of the four-current C normal to the four-velocity F.
Generally, for any pair of physical quaternions a, b, the expression

ba>

represents that part of the four-vector corresponding to a, which
is normal to the four-vector b (Note i). The above statement
is deduced from this, remembering that 1Y=ic.

* Of which the first and the last, denoted for subsequent reference by 77 and f,
are the quaternionic equivalents of Minkowski's world-vectors of the first kind
4> and ^, called by him elektrische Ruh-Kraft and magnetische Ruh- Kraft
respectively. Cf. his Grundgleichungen, pp. 33-34.

t Minkowski's matrix-form of the above relations is

where F is the matrix corresponding to the quaternion Y, and the remaining
symbols are as in footnote on p. 264. In these formulae we have put, after
Minkowski, c—i.

PONDERABLE MEDIA 267

In the course of the above calculations we came across the formula
= p- (Iv)/V2. Its inversion will be

Substituting here (IV) = y (Iv) - ypv2 and remembering that I = 3E 4- />v,
we obtain the interesting relation

about which a few words will be said later on. To resume the
above results :

The equations for a moving isotropic* conducting dielectric,
obtained from Maxwell's equations for stationary media, are in-
variant with respect to the Lorentz transformation. They consist
i° of a set of differential equations not containing the velocity of
motion at all, and 2° of a set of relations concerning the substantial
properties of the medium and involving its velocity v relative to
the observing system. The quaternionic form of these two sets of
equations is given in (la) and (2*3), where |8p, L are left-handed
and J5» B right-handed physical bivectors, and C a physical
quaternion, ~ q. The vector form of the first set is

-~-

= c. curl M ; div (£ = p

and that of the second set

where Ex stands for E + - Vvjft, etc., as in (A), and K, p, a- for

the permittivity, inductivity and conductivity of the body, as
originally defined from the standpoint of the rest-system.

* If K, etc., were vector operators, the passage from (2) to (2a), via (B), would
not be legitimate. In fact, ((Exv) would then be equal to(ATix.v), which has
nothing to do with A^(Exv), the former expression being a scalar and the latter
an operator. It is for this reason only that we have limited ourselves to isotropic
bodies. The case of anisotropy has not, to my knowledge, yet been treated, and
may be left for the reader's own investigation.

268 THE THEORY OF RELATIVITY

These are Minkowski's equations. They were first given in his
fundamental paper of 1907, in both their vectorial and matricular
forms already quoted. We may notice here that Minkowski himself
assumed that Maxwell's equations (i') and (2') are valid (in the
corresponding instantaneous rest-system S') at each point of the
material body, whatever the state of motion around that point, just
as if the whole body were fixed in S'. It is this that he calls his
' first axiom ' (loc. tit., § 8). Such being Minkowski's starting-point,
he asserts, consequently, the validity of the resulting equations (i)
and (2) for each element of a material medium moving in an arbitrary
manner with respect to the framework S, in short, for v varying in
both space and time. His only restriction is that v<c. Now, it is
not unlikely that the first set of Minkowski's equations can claim
such a general validity. (Notice that these are, properly speaking,
two equations for five vectors, otherwise yet unconnected.) But the
case is different when the first set is supplemented by the second.
For, apart from other reasons, if we pass to K= ^=i and a- = o, the
whole of equations (i), (2) reduce, as will be seen presently, to
the vacuum-equations, and the acceptance of the latter for frame-
works whose relative motion is variable, would require a thorough
reconstruction of the principle of relativity underlying the whole
theory. Retaining, therefore, this principle, we can consider Min-
kowski's equations as rigorously valid only for uniform motion.
Accordingly our v has been treated from the outset as a constant
vector and Y as a constant velocity-quaternion belonging to the
body as a whole. Of course, as an approximation of more than
sufficient accuracy, the equations (i) and (2) can well be used for
velocities experiencing all such time- and space-variations as are
practically realizable. Thus, for instance, they can safely be applied
to bodies kept rotating, as in the case of Wilson's experiment ; the
unequal FitzGerald-Lorentz contraction and the ensuing stress with
its influence upon K, etc., being of the order of /3'2.

The comparison of the equations (i), (2) with those of Hertz-
Heaviside, Lorentz and Cohn, none of which satisfy rigorously
the principle of relativity, must be left to the reader. It is
given at sufficient length in Minkowski's paper. As to Hertz-
Heaviside's equations for moving bodies, we have already seen that
they are not even a first-order approximation to the observed
state of things, giving a full, instead of the Fresnelian, drag. In
fact, Hertz-Heaviside's equations are, by their very construction

VACUUM AS LIMITING CASE 269

invariant with respect to the Newtonian, and not to the Lorentz
transformation.

Let us now stop a while at Minkowski's equations in order to learn
some of their properties.

In the first place, if K= //. = i and o- = o, then Minkowski's equations
reduce at once to the fundamental or the vacuum-equations. In fact,
in this limiting case we have, by the third of (2), I = /ov, and if />' = o,
also p = o, by (3). Again, by the first and second of (2), (£X=EX
Hx = Mx i.e.

whence, by elimination,

and since /J=£i, <E = E, and similarly, ,|ft = M. Q.E.D. The same
result may be obtained from the quaternionic form (20). In the
present case |£ = L becomes identical with the electromagnetic
bivector of the preceding chapters. And since at the same time
Jl = R, the sum of the equations (la) gives at once Z>L=C.
Properly speaking, to obtain K=p= i, <r = o, we have (on the electro-
atomistic doctrine) to consider a region outside the electrons, or at
least outside electronic assemblages crowded within atomic regions.
Then p = o, Z>L = o, and here the macroscopic bivector coincides
with our previous microscopic L. Thus the announced reduction
becomes complete.

As regards the meaning of the vector I, we have already remarked
that it is the sum of the convection- and the conduction-current. In
virtue of the properties of the stretcher e, the longitudinal component
of the latter current will be

and the transversal ones

This is in explanation of the short form of the third of (2), which may
be looked at as the expression of Ohm's law. If, for instance, e~2Ex
is considered as the resultant E.M.F. per unit length, then i/o-y will
be the specific resistance for the ^standpoint. This is one simple

270 THE THEORY OF RELATIVITY

way of splitting the conduction current into factors. But since, thus
far, the only requirement is that * resistance ' should reduce to i/cr for
v = o, we may equally well give the name of ' electromotive force '
to the line-integral of the vector Ex itself; then we shall have the
specific resistance-operator e2/o-y, instead of an ordinary scalar.
If second-order terms are neglected, the distinction disappears. The
conduction current may then be written, with more than sufficient
approximation,

I = o-Ex.

We will not stop here to discuss the nomenclature proposed by
various authors for Ex and its magnetic companion. It seems
advisable to leave them for the time being without any names.

The integral properties of Ex and Mx, in relation to JE, etc., may
at once be put into a form with which the reader has become
familiar in Chapter II. In fact, by (A) and (i), we have

-c. curl Ex = - c. curl E - curl VvJE

= ^ + v div JE + curl V JEv,

and this is precisely what in Note 2 to Chap. II. has been called

current (Jft).

That is to say, if d<r be a surface element composed always of the
same particles of the body, and n the normal of db-, we have

<r(n . curl Ex) = - —
Similarly,

f(n . curl Mx) = ^ (to da-) + (In).

Recurring, therefore, to Stokes' theorem, we have for any surface <r,
which together with its bounding circuit s is carried along with the

body,

r T r T ^ r

:n)dcr, (4)

(5)

These are the required formulae. Returning to p. 23 (where
Maxwell's law I. is to be supplemented by the conduction current)

BOUNDARY CONDITIONS 271

and to p. 30, the reader will find this integral form of equations
most suitable for a direct comparison of Hertz's theory with
that of Minkowski. Instead of Hertz's E, M we have here Ex,
Mx, and instead of his (£ = .AT3, etc., the Minkowskian relations
(2), involving the velocity of the medium relative to the observing
system.

Applying (4) to a pair of surfaces bounded by one and the same
circuit 5, as on p. 25, we obtain the familiar equation,

(6)

where e is the total charge of any portion of the medium enclosed
completely by the surface o-, whose outward normal is n. If the
bounding surface is entirely composed of lines of conduction-current,
then the charge remains constant. The same result follows, of
course, from the first pair of the differential equations (i), with
I = />v + 3E. And since these are independent of the Minkowskian
connexions, involving the substantial properties of the medium,
there is no wonder that the equation of continuity reappears in its
familiar form.

The above equations (4) and (5) lead at once to a pair of what are
usually called the boundary conditions. The other pair follows
directly from div (& — p and div Jft = o. In fact, let ^ be, in Hada-
mard's phraseology, a stationary surface of discontinuity,* i.e.
permanently affecting the same material particles, such as the surface
of contact of two different media. And let us require that I and the
individual time-rate of change of (j£ and Jft should ^finite. This
condition, to be fulfilled at any point of 2 and elsewhere, is necessary
to prevent <£, ffi mounting up to infinite values at any point of the
medium.! Under these assumptions apply (4) and (5), in the usual
way, to an infinitesimal rectangle, with its shorter sides normal to 2.
Then the result will be that the tangential components of Ex and Mx
must be continuous. The two remaining conditions are as in the
older theory. They follow at once from the divergence-formulae,
and require the normal component of Jft to be continuous, and the

*To be carefully distinguished from a wave of discontinuity, which is pro-
pagated in the material medium. The reader unfamiliar with this subject is
referred to the author's Vectorial Mechanics, pp. 128 et seq.

f While it is not necessary at all for a wave, whose singularities do not remain at
the same particles, but pass by and are transferred to others and others.

272 THE THEORY OF RELATIVITY

jump of the normal component of OB to be equal to the surface-
density of charge. Thus, if there is no such charge, we have the
following boundary conditions :

(Jftn) and (OBn) continuous,
VnVExn and VnVMxn continuous,

where n is normal to the boundary. The latter pair of expressions
gives the tangential parts of the vectors, i.e. in both size and
direction.

Next, as regards the formula (3) for the density of charge, which is
a consequence of the nature of C as a physical quaternion. Suppose,
first, that there is no conductivity. Then

just as for the microscopic density of charge, whence, for any portion
of the body,

l/o ^5"= |/oV,S" = £',

which means relativistic invariance of macroscopic charge. This
property then continues to hold for a moving body, provided that it
is a perfect insulator.

On the other hand, suppose that the body is conductive, but that
there is no rest-charge (p = o). Then there will be for the ^-observers
an apparent charge of density

/•-$<*»)• (8)

The history of this conduction charge, or compensation charge, as
it previously has been called, can be traced back as far as 1880,
in which year it was deduced by Budde ( Wied. Ann., Vol. X.
p. 553) from Clausius' fundamental law of electrodynamics. Budde,
whose formula differed from the above one by containing unity
instead of y2, was able to defend Clausius' law from a serious attack
by showing that this charge accounted for the non-existence of an
action between a current circuit and a charged body sharing in
the earth's motion. Hence the name of 'compensation charge.'
In 1895 Lorentz, by averaging his electronic equations, obtained
for the density of this charge a formula which was wholly identical
with (8). See § 25 of his Essay. A careful comparison of the

WILSON-EFFECT 273

two ways leading to one and the same result will be found useful,
and the electronic interpretation of a formula which here appears
as a relativistic consequence of Maxwell's equations will not be
lacking in interest. But even apart from electro-atomistic con-
cepts the reader will not fail to see that if the densities of positive
electricity, flowing one way, and negative flowing the other way,
cancel one another for an observer attached to the conducting
body, then the corresponding values p+ and /o_, as estimated from
any other (S-) point of view, will in general not annul themselves.
They will do so only when the current has no longitudinal com-
ponent. There is no difficulty in working out the quantitative
details of such a reasoning, and thus re-obtaining the above formula.

Next, as regards the dragging of waves. We know already from
Chapter VI. that, whatever the value to' of the velocity of propagation
in the rest-system, its lvalue to will follow by the addition theorem
of velocities, and will give, therefore, the Fresnelian coefficient.
And that Einstein's theorem is in fact applicable to the present
case, can be concluded from the manner in which the equations
(i), (2) have been obtained from those, (i'), (2'), holding in S'.
Thus we know beforehand that Minkowski's equations will lead
to the correct Fresnelian value of the dragging coefficient. And
this expectation is readily confirmed on performing the explicit
calculation. Cf. Note 2.

Finally, let us remark that Minkowski's electromagnetic equations
account fully for the well-known results of Rowland's, Wilson's,
Rontgen's and Eichenwald's experiments. We cannot enter here
upon the corresponding details, and must confine ourselves to
short indications concerning each of these famous experiments.
The magnetic effect of the cotivection current, first proved experi-
mentally by Rowland, and confirmed by other physicists,* is
directly expressed by the term /ov, which together with the
conduction current makes up I, and thus equally with that current
contributes to the magnetic field. It is scarcely necessary to say
that the Rowland effect was equally well expressed by the Hertz-
Heaviside equations. The result of Wilson's experiments on the

* H. A. Rowland, Amer. Jonrn. of Science, Vol. XV. 1878, p. 30. H. A.
Rowland and C. T. Hutchinson, Phil. Mag., Vol. XXVII. 1889, p. 445.
H. Fender, Phil. Mag., Vol. II. 1901, p. 179. E. P. Adams, ibidem, p. 285.
II. Fender and V. Cremieu, Comptes rendtis, Vol. CXXXVI. 1903, pp. 548,
955. A. Eichenwald, Ann. der Physik, Vol. XI. 1903, p. I.
S.R. S

274 THE THEORY OF RELATIVITY

electric effect of rotating a dielectric between the connected plates
of a condenser in a magnetic field M consisted in each of the
plates being found charged to a surface-density

(K-i)l*M (Wilson)

of opposite signs.* In the theoretical treatment of the problem
uniform translation (of each element) can, with sufficient accuracy,
be substituted for the actual spin, and the state being supposed
stationary (and <r = o, /o = o), Minkowski's differential equations reduce
to curlE = o, etc. Using these, with the appropriate boundary-
conditions, and the first pair of (2), Einstein and Laub f deduce,
for the surface-density in question, the value

(Mnk)

with the correct sign for each plate. The authors observe that
Lorentz's theory would give, instead of this,

(Lor)

Since in Wilson's case ft was = i, both of these theoretical formulae
coincide with his experimental result. If a dielectric of considerable
inductivity were available, experiment would readily decide in favour
of the former or the latter theory. As to Hertz-Heaviside's theory,
it would give for the Wilson-effect

A'pftM, (HH)

i.e. practically KfiM, which is equally contradicted by Wilson's and
by Blondlot's results. This disagreement, even in the case of a
first-order effect, might have been expected, in view of the fact that
Hertz-Heaviside's equations give a full instead of a Fresnelian drag.
Lastly, as regards the experiments on the magnetic effect of moving
polarized dielectrics, which were first carried out by Rontgen
and more recently with increased accuracy by Eichenwald, { it will
be enough to write down the expression of what is generally called

*H. A. Wilson, Phil. Trans., Vol. CCIV. A, 1904, p. 121. Wilson's positive
result agrees with the absence of any such effect stated previously by R. Blondlot,
Comptes rendits, Vol. CXXXIII. 1901, p. 778, in the case of air as dielectric,
for which A" differs but little from unity.

tA. Einstein and J. Laub, Ann. der Physik, XXVI. 1908, p. 532.

JW. C. Rontgen, BerL Sitzungsberichte, 1885, p. 195; Wied. Ann.,
Vol. XXXV. 1888, p. 264, and Vol. XL. 1890, p. 93. A. Eichenwald, Ann.
der Physih, Vol. XL 1903, p. 421.

ROENTGEN-CURRENT 275

the Rontgen-current. In fact, if we limit ourselves to homogeneous
media, the experimental results may be concisely stated by saying
that the observed value of the Rontgen-current is

(A'-i)curlVEv. (Exper.)

Now, according to the Hertz-Heaviside equations (p. 31), this
current would be

A'.curlVEv, (HH)

so that the disagreement is exactly of the same kind as for the
Wilson-effect. On the other hand, Minkowski's equations, with
/*= i, give for the Rontgen-current the rigorous value

curlV[(£-B]v, (Mnk)

where, by the first of (2) and by (A),

Thus the first-order term of the Minkowskian expression represents
correctly the observed facts. The second-order terms are, of course,
for the time being far too small to be detected. The Minkowskian
value of the Rontgen-current follows also from a later form of
Lorentz's equations deduced (1902) from the electron theory.* In
what consists the violation done by these last equations to the
principle of relativity may be seen from Minkowski's paper. There
the reader will find also the appropriate coordination of the field-
vectors involved in the various theories.

So much as regards the electromagnetic equations for moving
bodies, contained in (i) and (2). Now for the dynamical part of
the subject. Before proceeding to a relativistic construction of the
formulae for the ponderomotive force and the associated physical
magnitudes, some preliminary remarks seem indispensable. These
will concern the requirements to be postulated in addition to those
dictated by the principle of relativity itself. The choice of such
supplementary requirements or postulates is free, within fairly wide
limits. We shall select those which seem to offer the advantage of
possible simplicity and which will lead to results but slightly different
from the ponderomotive formulae originally proposed by Minkowski.

* Amsterdam Proceedings, 1902-1903, p. 254. See Lorentz's article in EncykL
der math. Wiss., Vol. V2. pp. 208-211, and in particular formula (XXVIL), in
which |3 = £p - <g corresponds to the above <E - E.

276 THE THEORY OF RELATIVITY

Let P be the ponderomotive force due to the electromagnetic field,
per unit volume of the medium, and, therefore, (Pv) its activity.
Further, lety ^Joule's heat, or the Joulean waste, per unit time and
unit volume, and ^the force-quaternion, i.e., according to what has
been said in the last chapter,

^= •? [(Pv) +/] + !>• (9)

Let u be the density of electromagnetic energy, g that of electro-
magnetic momentum, and finally / and jp the ('absolute,' not
relative) stress-operator and flux of energy, as defined in the usual
way with respect to the observing system S. With this meaning of
the symbols, let our requirements be as follows :
i°. P, a physical quaternion,

^[(Pv)+/] + P-'/. (a)

2°. Principle of momentum, to call it by its usual short name, that
is to say,

where V/ stands for 3fi/3# + 3f2/3y + 3f3/9s.
3°. Principle of conservation of ejiergy, i.e.

(Pv)+/=-g-divl3, (y)

where $ has, thus far, nothing to do with the momentum.

It is needless to add that, besides fulfilling these explicit require-
ments, the resulting formulae have to agree with experience, as
far as it goes, and to reduce, for A'=/x=i, cr = o, to the previous
vacuum-formulae, as, in fact, they will.

We have seen in the preceding, chapter that there is at the present
time a strong tendency to universalize the simple relation of equality
holding between g and J3/V2 in the ideal limiting case of a vacuum.*

*This tendency was initiated by Planck's paper (Phys. Zeitschrift, Vol. IX.
1908, p. 828) on the principle of action and reaction. M. Abraham uses the
equality r2g = $ throughout his papers (quoted on p. 240), putting it at the base
of his electrodynamics of moving bodies, which is also adopted in Laue's Rela-
tivitatsprinzip. That equality is called by Laue ' the theorem of the inertia of
energy,' and plays in his book the part of an universally valid relation. But his
own way of introducing this 'theorem' (p. 164 of the 2nd ed.) will show best
how vague are the reasons for accepting it without limitation.

POXDEROMOTIVE FORCE

277

But, as far as I can see, there is nothing to compel us to such a
generalization. If it is assumed that the matrix embodying the
stress, momentum, etc., should be symmetrical, then, of course,
the equality under consideration follows from (ft) and (y). But
nothing prevents us from abandoning, at least in the case of ponder-
able media, that assumption of symmetry.* We shall see that in
doing so we need not even give up the formulae (14) or (140) of
Chap. IX., which have led to so many far-reaching consequences.
These formulae will continue to hold within wide limits, although the
more general formula (10) of that chapter will have to be modified.
Thus, there will still be * inertia of energy,' with its manifold
corollaries.

So much to justify the abandoning of the assumption of universal
proportionality of momentum and energy-flux.

Returning to our above requirements, let us, first of all, observe
that, with the given meaning of F, assumptions (/3) and (y) may be
condensed into

^=-lorS, (10)

where

or, written out fully,

./22» -/23 >
/32' /33'

(llfl)

*In Sommerfeld's four-dimensional algebra (loc. cit.), the symmetrical world-
tensor, corresponding to such a matrix, is generated by what he calls ' a
complete multiplication ' of a six-vector into itself. But why not multiply two
different six-vectors ' completely ' into one another ? Such a procedure is
exemplified, in matrix-form, in Minkowski's paper. But, apart from the process
of generation, any given matrix of 4x4 constituents can be used for
relativistic purposes, provided that its product into a four-vector (matrix) gives
again a four-vector.

278 THE THEORY OF RELATIVITY

Here, in general, fix ^fai, so that the matrix lacks symmetry
altogether.

Next, to satisfy (oc), we have to write, for any pair of legitimate
frameworks of reference 6" and 6", as on p. 239,

§ = A§'4, (12)

where A, A are as before. This fixes the transformational properties
of the stress, momentum, etc., quite independently of the electro-
magnetic expressions they will hereafter receive. Developing (12),
we have the following table of Cartesian formulae, which take
the place of (loa), p. 237, and which, though not needed for
our electrodynamical investigation, are here given because of their
bearing upon the subjects treated in the preceding chapter :

; /22=/22'; /33=/33'

- (l2d

/32 =/M ; /IS = 7 (/is' + *£s ) ; /21 '

Here the ^c-axis is taken along v, the velocity of S' relative to S.
(The reader can condense these formulae into a more convenient
shape by using vectors and the stretcher e.) If there is, from the
S'-point of view, no flux of energy and no momentum, then u and
the stress-components become as in (14(1) of Chap. IX. ; we
obtain also the same *S-momentum as before, i.e.

whereas

Thus, fig and ^9 may still differ from one another. But if the
stress in S' is self-conjugate, the two vectors become equal, and the
formulae of Chap. IX. are again obtained. In the case of
electrodynamics, for instance, the latter condition, /IK' =/*/, will be

PONDEROMOTIVE FORCE 279

seen to hold for any electromagnetic field, if S' is attached to the
ponderable medium ; and the condition of vanishing g' and $' will
be satisfied in the case of a purely electrostatic, or a purely
magnetostatic field.

With /t<' =/«/ alone, we have, from (120), the interesting relation

3-*t-;[!P'-<V], ('3)

which will hold for any electromagnetic field, provided that S' is
the rest-system of the ponderable medium.

But let us return to our chief subject. After what has been
said we could either employ the form (10) of the force-quaternion,
and would then have to prove that § is transformed according
to (12), or we can proceed by satisfying our three requirements
in their original forms (/3), (y), and (oc). The two ways are wholly
equivalent to one another. Minkowski chooses the former : he
constructs § in a manner that ensures by itself the validity of
(12), subjects it to the operation lor, and develops the resulting
four-vector.* We shall take the latter way, which the reader may
find easier to follow. Thus, we shall first construct F so that it
should be a physical quaternion, and then find the corresponding
expressions for the energy, stress, etc., according to (/3), (y), aided,
of course, by the electromagnetic equations (i), (2).

The first step to be taken is suggested by analogy with the
construction of the fundamental electronic force-expression (cf.
p. 220). We know that

and that L = Jtt - iE is a left-handed bivector. Therefore, CL ~ q.
Similarly, E = Jrl + tE being a right-handed bivector, we have
R C ~ q. The difference of both products has also the structure
of q, and thus is again a physical quaternion, and can be used as
far as (oc) is concerned. Try, therefore, to satisfy the remaining
requirements of the problem by putting

J?=l[CL-'RC]. (14*)

This will turn out to represent the whole force-quaternion in the
case of a homogeneous medium, and will, for heterogeneous media,

* See Note 3 at the end of the chapter.

28o THE THEORY OF RELATIVITY

be easily supplemented by another physical quaternion involving
the variations of A", //. Develop the right-hand side of (140).
Then the vector part will give the ponderomotive force,

(15*)

and the scalar part will lead to

(i6a)

Eliminate (El) from these two equations and remember that
Then the result will be

giving for Joule's heat the expression

7-(BBx). (-7)

Thus far (ft) and (y) have not yet been employed. Now take
account of these conditions, beginning with the latter. This gives,
by (i6a),

Now, by the electromagnetic differential equations (i),
- (El) .1 ~ (E$ + MJH) + c . div VEM.

Thus (y), the principle of conservation of energy, is satisfied if the
density of electromagnetic energy is taken to be

^ = i(E<£ + Mjl), (18)

and t\\eflux of energy, from the standpoint of the observing system,*

$ = *VEM. (19)

The addition of an arbitrary sourceless (solenoidal) flux, as well as of
an invariable w-term, would be irrelevant.

Lastly, to represent the ponderomotive force in the form required
by (ft), introduce in (150) the first pair of equations (i). Then

* Remember that 3/9/ is the symbol of local time-variation in the observing
system S.

ELECTROMAGNETIC STRESS 281

Using the second pair of equations (i), and writing, for the moment,

we have

This gives, first of all, for the electromagnetic momentum per unit
volume,

(20)

and what remains to be shown is that the vector sum A, familiar from
the Maxwellian theory, is of the form - Vf. Now, this is exactly the
case, provided that K and /x, involved in (2), are constant throughout
the medium. In fact, take for the electromagnetic stress the familiar
expression

fM = im - E(<En) - M(|ttn), (21)

where u is as in (18). Then, remembering that Vf is used as
shorthand for 'dtlfdx

On the other hand, we have

£)-(<£. V)E

(where the dot stops V's differentiating action), and a similar ex-
pression for the last term of A. Thus,

-V/=A + N,
where

N

To prevent a possible misunderstanding, we may add that this is a
vector whose components are

SM 3< 9

etc.

Now, returning to the relations (Bx = ^Ex, JEX =ftMx, and effecting
a transformation, the details of which will be found in Minkowski's

282 THE THEORY OF RELATIVITY

paper,* the reader will verify that the above vector is identical with
N = - i (Tr/)2 . VAT- i(T£)2 . V/x,

where the quaternions v and £ are as in (B), p. 265. In the case of
homogeneity, therefore, N vanishes, and we have A = - V/j so that
the condition (/3) is satisfied, with the above stress and momentum,
by taking (15^) for the ponderomotive force, that is (140) for the
force-quaternion.

In the more general case of a heterogeneous medium we have only
to supplement our original P by the vector N, and consequently to
add to our original F the quaternion

(c)

which, like that F itself, is ~ ^, since Tr) and T£, being the tensors
of physical quaternions, are invariant with respect to the Lorentz
transformation.

Thus we shall have, as a generalization of

f 1 F /^T ^y S~* ~\ 1 / T1 M \ ^ 7~} "L^

./'= £[Cli — ICC J — £( 1^)". DK —
which splits into

c
and

?*&' ^)//.

n- (l6)

All requirements being now satisfied, with the above values of
density and flux of energy, and of stress and momentum, the only
thing to be still revised on account of the heterogeneity of the
medium is the Joulean waste. Now, proceeding as before, we
obtain at once, from (15) and (16),

7=(£EX) + 1(^)2^

where

dK

and similarly, <//*/*# =a/A/y3/' '. Thus, if there is, from the stand-
point of the rest-system, no time-variation of K, ^ we re-obtain

Grundgleichungen, formula (92), in which the last term, due to the changes of
the velocity of motion, is to be omitted.

PONDEROMOTIVE FORCE, ETC. 283

the previous value, (17), of Joule's heat. Under these circumstances
we have also

which values can be substituted in the last two terms of (15).
To resume :

The force-quaternion

F=$[CL-*C] - i(Tr,)2. DK- J(TC)2. D^ (14)

being a physical quaternion, satisfies the fundamental relativistic
requirements and, at the same time, the so-called principle of
momentum,

and the principle of conservation of energy,

o. (y)

It gives for the ponder omotive force, per unit volume of an isotropic
medium,

V/., (15)

and for the Joulean waste (with "bKfdf = "dpfdf = o) :

/=(3EBX), ('7)

where 3E = I-/ov is the conduction current. The corresponding
auxiliary magnitudes are as follows :
The density of electromagnetic energy

(18)

the flux of energy

(19)

the density of electromagnetic momentum

g=^voytt, (20)

and, finally, the electromagnetic stress

fn = wn - E(£n) - M(|ttn). (21)

The physical quaternions // and ^ are, as on p. 265,

(22)

284 THE THEORY OF RELATIVITY

Our ^is the quaternionic equivalent of Minkovvski's four- vector K
(loc. cif., §14, for constant v, of course), and so also our stress-
components, etc., are identical with the sixteen constituents of
Minkowski's matrix - S. The difference between the theory here
proposed and that given by Minkowski, is this : — What Minkowski
considers as the ponderomotive force is the vector, ?wt of F itself
but, of

i.e. of that part of the four-vector F, which is normal to the four-
velocity Y. Thus Minkowski's ponderomotive force is not of
the form (/3), though it becomes so in the rest-system. The reason
why Minkowski proposed for the four-force the above part of
F, instead of the whole F, is to be sought for in his dynamics,
according to which the 'moving5 four-force had always to be
normal to the particle's world-line. This corresponded to the
assumption of a constant rest-mass. But in general, as has been
explained in Chapter IX., the four-force does not necessarily bear
that relation to the world-line, and it will not do so whenever
there is heat supply or heat generation. Now, this being exactly
the case with an electric conductor, we had to abandon the
Minkowskian condition of orthogonality. And, in connexion with
this, Joule's heat has, from the beginning, been embodied into our
force-quaternion along with the activity of the force, such a pro-
cedure being directly suggested by the dynamical considerations
of Chapter IX. If there is no conductivity, and therefore also no
Joulean waste, then the four-force represented by (14) is, in fact,
normal to the world-line of the body, i.e.

S YCF= o, for cr = o,

as is proved in Note 4 at the end of the chapter. But in a con-
ducting body this property does not hold.

We will content ourselves here with the above modification of
Minkowskian electrodynamics, which, besides fulfilling the stated
requirements, is also in complete agreement with what is known
from experience. It is true that Einstein and Laub * missed in
the ponderomotive force of the above kind a term due to dis-
placement-current (uniform with those due to conduction- and

* Ann. der Physik, Vol. XXVI., 1908, p. 541.

PONDEROMOTIVE FORCE, ETC. 285

convection-current, contained in VIJH/<r). But, since nobody has
ever observed such an action of the magnetic field, this can hardly
be considered as a serious objection. In order to obtain the
desired term, Einstein and Laub recurred to the electron theory,
and since in doing so they thought necessary to limit themselves
to the case of stationary bodies, their electrodynamics is of no
particular interest from the relativistic point of view. Abraham's
ponderomotive force contains, in addition to (15), several other
terms, and among these the displacement-current term. His electro-
dynamics of moving bodies,* as has already been mentioned, is
based upon the assumption of the particular relation g = J3/^2,
borrowed from the vacuum-equations. The desire to retain this
relation throughout the theory makes Abraham's formulae con-
siderably more complicated than the above ones. His expression
for the Joulean waste, subject to the same conditions for K, /*,
is the same as above, but those for the density and flux of energy,
and consequently also for the momentum, are less simple.

The set of electrodynamical formulae given above is best charac-
terized by saying that, while satisfying the principle of relativity
and the principle of conservation of energy, it gives for the rest-
system the ponderomotive force

(23)

In fact,/ as given by (21), reduces in S to the Maxwellian stress-
operator,

f=f\^ = \(KE'* + v.M"*)-KV(F -/xM'(M' .

It will be remembered that Maxwell's stress, taken by itself, would
give, in absence of electric charge and of ponderable matter, the
ponderomotive force

and this ' force on the free aether ' is just balanced by the second
term in (23). See p. 48. With the exception of this obviously
desirable supplementary term everything is as in Maxwell's electro-
dynamics of stationary bodies. Thus, (15), the developed form
of the ponderomotive force, becomes in the rest-system, by (A),

F = p'E' + - V£ffl' - \E*. VK- J J/'
* Cf. footnote on p. 240.

286 THE THEORY OF RELATIVITY

where 3t' = T is the conduction current, and each of the four terms
has its old familiar form and meaning. Again, by (17) and (A),
the Joulean waste takes its usual form,

while (18) and (19) give at once the Maxwellian density of electro-
magnetic energy, and the familiar Poynting vector for the flux
of energy,

The transformation formula for Joule's heat is easily obtained.
In fact, since F, denned by (9), is a physical quaternion, and since

we have at once, writing Pl for the longitudinal component of
the force,

and

whence, by subtraction,

/=y(i-^)7 = i/'.

Consequently, if </,S' be any volume-element of the body, and
^6" its correspondent,

in complete agreement with (29), Chap. IX.

The electromagnetic momentum bears, in the rest-system, a
simple relation to the energy flux. In fact, by (20),

g' = ^ VE'M' = :4 VE'M',

c to z

where, dispersion being disregarded, to' is the velocity of propagation
of disturbances, as estimated by the ^"-observers. Hence, instead
of Planck's relation, we have

g' = p|l', (*4>

so that, in a stationary ponderable medium, to' takes the place of c.
And since to' plays in such a medium just the same part as the

ENERGY FLUX AND MOMENTUM 287

critical velocity in empty space, it seems quite natural that (24)
should replace the relation which holds good in the absence of
matter. The stress in 6" being self-conjugate, our previous equation
(13) can be applied, so that, in general,

where n is the refractive index of the medium. If, therefore, n
differs at all from unity, we have J3=/=<r2g, unless there is in the
rest-system no Poynting flux.

Finally, notice that if K=p=i, and o- = o, the ponderomotive
force (15) coincides with that of the electron theory. And the
same thing is true of the above expressions for stress, density and
flux of energy, and momentum. So also were the vacuum-equations
contained, as a limiting case, in Minkowski's electromagnetic differ-
ential equations for moving bodies.

NOTES TO CHAPTER X.

Note 1 (to page 266). Let the quaternions a and b represent a pair of
four-vectors. Then the component of the four-vector a taken along
the four- vector •£ (cf. p. 148) will be represented by

and, consequently, the part of a normal to £, in both size and direction,
by

Now, S<2c^ = i[^ctf + #c^]» and #«— (T£)*. Hence

baj>*

which is the required expression.

Note 2 (to page 273). It will be enough to consider here the case of
plane waves, propagated along v, in a non-conducting medium, carrying
no charge, so that I = o.

As in a previous Note (p. 59), take E, <E, etc., proportional to an
exponential function of the argument

where g is an imaginary constant, and b the velocity of propagation,
as estimated from the 5-point of view (or else consider a wave of

288 THE THEORY OF RELATIVITY

discontinuity). Then the equations (i) and (2) will give, by (A), p. 265,
and since v=i

the solenoidal conditions (<8i)=(Jfti)*=Q being already satisfied by (a).
Next, introduce (a) into (b\ (c) ; then

showing that E and M are again transversal. Use the latter relations
in (a\ eliminate either E or M, and remember that

b'
Then the result will be

*

i-
whence

i + •z/b'/if2

Thus to is obtained from to' and v by Einstein's addition theorem of
velocities, and this, as we saw on p. 172, gives the Fresnelian value
for the dragging coefficient.

Note 3 (to page 279). Let h and //, as on p. 264, be the alternating
matrices equivalent to the electromagnetic bivectors IP and L respec-
tively, i.e.

^31 = ^2,

and

14= - l\ , 24 = ~ 12 > 34 = ~ 3 .

Both of these matrices reduce, for A^=^/x=i, to the matrix h of Note 2
to Chap. IX.

Minkowski begins by constructing the_product of h into H. Since
each of the factors is transformed by A( }A, the same will be true
of their product, which will be a matrix of 4x4 constituents. Now
similarly as on p. 259, the reader will find

-hff=g + \, (a)

where <S is the matrix of the present chapter, whose constituents are
exactly those given by (18) till (21), and

(b)

STRESS-ENERGY MATRIX 289

or what in the rest-system becomes 'the Lagrangian function.' (It may
be mentioned, for the sake of comparison with Minkowski's paper, that
our //, H, A, <S, F are his /, Ft L, — S, K respectively.) Similarly,
h* and H* being the dual matrices,

-ff*A*=-g + X. (c)

By (#) and (c\ A' is an invariant, and <S is transformed by A ( }A,
so that

is a genuine four-vector (or physical quaternion). The latter becomes,
by (a), (c} and (*),

F= lor h . H- lor H* . h* + N, (d)

where the dots act as separators and N is the four-vector written in
quaternionic form under (c), p. 282.

Next, using in (d] the differential equations of the field, lor h = - s
lor H*=o, Minkowski obtains

(e)

where s is the current-matrix, represented in this chapter by the
quaternion C. Minkowski's ponderomotive four-force is the part of (e)
normal to the four-velocity. Our force-quaternion (14) is the quaternionic
equivalent of the whole four-vector (e).

Notice that (£) can be written, in terms of the electromagnetic bivectors,

whence the invariance of A is seen immediately.

Note 4 (to page 284). For a non-conducting medium, the current
quaternion becomes

c cy
and therefore, the force-quaternion (14),

where 2t»y= KL - R K, as on p. .265. Hence,

Again, as regards the second term of F,

since 'dK[dt = o, by assumption. Similarly for the last term of the
force-quaternion. Thus S FcF=o, which was to be proved.

S.R.

INDEX

(The numbers refer to the pages.)

Aberration, 35, 38-39, 60, 62, 70

Abraham, 240, 276, 285

Absolute period, 69

Acceleration-quaternion, 185

Activity, of ponderomotive force,
238

Addition of quaternions, 152

Addition theorem of velocities, 126,
164

Aether, 17-18, 35, 37, 42, 43, 45,
62-63, 73, 87> 89, 99, 117

Aft-cone, 136, 138

Airy, 38, 39

Alphonso X., 4

Alternating matrices, 229, 259

Amplified systems, 13

Angle, of a quaternion, 154
of parallelism, 180

Angular momentum, 5

Anisotropy of rest-mass, 249

Antivariant quaternions, 199'

Arago, 36

Arrhenius, 43

Associativity of matrix products,

161

of quaternionic sums and pro-
ducts, 152, 153

Asymptotic cone, 136, 138

Atomism, electro-, 43

Averages, 51

Axial vectors, 146

Axis of a quaternion, 154

Behacker, 241
Bessel, 249

Biquaternions, 203, 219
Bivectors, 201

electromagnetic, 45, 209, 263
Blondlot, 274

Bolyai-Lobatchewsky's space, 176
Bonola, 177, 178
Born, 190, 260
Boscovich, 38, 60

Boundary conditions, 272
Brace, 83
Bradley, 34, 35
Budde, 272

Campbell, 39

Canal rays, 106

Cauchy's symbol, 113

Causality, 8

Cayley, 143, 156

Charge, 25-26, 31

an invariant, 207, 228

Chemical reactions, 256

Clausius' law of electrodynamics,
272

Clifford, 203

Clocks, moving, 106

Cohn, 26, 268

Commutativity of quaternionic sum,
152

Compatibility, kinematical condi-
tion of, 56

Compensation charge, 272

Complementary bivectors, 229
electromagnetic bivector, 218

Complete systems, 9, 18-20

Complex vectors, 200

Composition of velocities, 116, 126,
163-181

Condenser, 250

Conduction charge, 272
current, 265, 269

Conductivity, 261, 267

Cones, fore- and aft-, 136, 138

Conjugate diameters, 132, 139
quaternions, 152

Conservation of areas, 198
of energy, 233, 283

Contraction hypothesis, 78-83
relativistic, 105

Convection current, 31, 44, 265,

273
potential, 81, 213

INDEX

291

Convective fields, 211-215
Conway, 150
Coordinates, effective, 85
Copernican system, i, 4
Corpuscular theory of light, 35, 36,

60, 61, 73

Corresponding states, 68
Co variance, 145
Co variant quaternions, 158
Cremieu, 273
Current, displacement-, 22, 30, 44

in a moving medium, 59

magnetic, 22, 30

-quaternion, 46, 207 ; 263

Rontgen-, 275

Dalembertian, 113, 216
Darwin, 3
Debye, 77

Degrees of freedom, 9
Density of electric charge, 25-26
Determinant, of a matrix, 143, 162
Dielectric displacement, 23
polarization, Lorentz', 53
Direction cosines, four-dimensional,

148

Discontinuities, 56-58, 120
Disintegration, 106, 257
Dispersion, 54, 261
Displacement-current, 22, 30, 44
Doppler's law, 70
Dragging coefficient, 33-4!. 55. 60,

71, 172-174. 273, 288
Dual matrices, 230
Duhem, 38
Dyadic, 125
Dynamics, of a particle, 192-198,

246 et seq.

Earlier, essentially, denned, 142

Earth as time-keeper, 6

Earth's motion, 17-18, 35-39, 60-62,

71
Effective coordinates, 85

time, 85

Eichenwald, 32, 273, 274
Einstein, 21, 87, 92, 94, 99, 164,

193. 195. 24?

Einstein and Laub, 260, 274, 285
Electric charge, 25-6, 31

force, ponderomotive, per unit

charge, 44, 81
Electrical moment, 52
Electromagnetic bivectors, 45, 263

discontinuities, 56-58

energy, 47, 233, 280

masses, 43, 215, 250

Electromagnetic momentum, 50,
234, 281

stress, 48, 234, 281
Electrons, 43, 52,56, 79, 197, 214-216
Elster and Geitel, 43
Energy, 245-251, 256-257

electromagnetic, 47, 233, 280

kinetic, 195, 255
Enthalpy, 251
Entropy, an invariant, 253
Eotvos, 249

Equation of continuity, 206, 216
Essentially incomplete system, 10

Faraday tubes, 23, 27, 250
Field, convective, 211-215
FitzGerald, 77, 78
FitzGerald-Lorentz hypothesis, 78
Fixed aether, 38
Fixed-stars system of reference,

5, 6, 17
Fizeau, 40

Fizeau's experiment, 40-41, 70
Flux of energy, 47, 233, 280
Force, Newtonian and Minkow-

skian, 193

Force-quaternion, 220, 254, 283
Fore-cone, 136, 138
Foucault, 3
Four-dimensional rotation, 127-128

vector algebra, 146-150
Four- potential, 217
Four-vectors, 140, 147
Four- velocity, 184
Fourier's equation, 20
Framework of reference, 2-6, 17, 1 8
Free aether, 43, 48, 73

electrons, 43

Fresnel, 36-39, 41, 60-62, 73, 99
Fresnel's dragging coefficient, 34,

37, 39, 41, 55, 71, 174, 288

Galilean transformation, 17
Geodesic, Lobatchewskyan, 178
Gibbs, 167, 222, 258
Giese, 43

Gravitation, 241, 249
Group of transformations, in general,
18

Lorentz-, 16, 167, 170

Newtonian, 17

of translations, 129
Gyroscope, 5

Hadamard, 57, 271
Hamilton, 203, 209, 151, 153

292

INDEX

Hamilton's principle, 225

Heat function for constant pressure,

251

Heat supply, 253
Heaviside, 24, 31, 34, 48, 222
Heaviside's ellipsoids, 214
Helmholtz, 43, 48
Hercules, constellation, 17
Herglotz, 247
Hertz, 30
Hertz-Heaviside equations, 24, 31,

4i, 275

Heterogeneous media, 282 et seq.
Huygens construction, for moving

mirror, 89-91

Hyperbola of curvature, 191
Hyperbolic functions, 133
motion, 190
space, 176

Hyperboloids, Minkowskian, 138
Hypervelocity, 114

Idemfactor, 124
Identical conditions, 56, 120
Index of refraction, 54
Individual variation, 31
Induction, magnetic, 23
Inductivity (permeability), 24, 261,

267

Inertia of energy, 247, 276
Inertial system of reference, 5, 17
Intermediate region, 142
Invariable plane, 5
Invariant, relativistic, 112
Isopiestic motion, 252, 254
Isotropic pressure, 245
Isotropy, of light-propagation, 28

Johnstone Stoney, 43
Joule's heat, 276, 280

Kaufmann, 181
Kepler's second law, 198
Kinematic conditions of compati-
bility, 56, 120

space, old and new, 176
Kinetic energy, 195, 255

time, 13
Kohl, 77

Lagrange, 5

Lagrangian function, 209, 225-227,

259, 289
Laplace, 5
Larmor, 43, 45

Latent energy, 257

Later, essentially, denned, 142

Laub, 77

Laue, 77, 106, 119, 237, 240, 245,

251, 276

Left-handed bivectors, 201
Length, of a four- vector, 148
Lengths compared, 101-107
Lie, 1 8, 19, 125
Lines of states, 19
Lobatchewsky, 177, 181
Lobatchewskyan rotation, 128
Local change, 27

time, 66, 84
Lodge, 77
Longitudinal discontinuity, 57

mass, 196, 214, 255

stretcher, 125
lor, matrix, denned and transformed,

145
Lorentz, 42 et seq., 193, 226, 261,

272, 275

Lorentz's equations, 44, 53
Lorentz transformation, Cartesian

form, 86, no
matrix form, 144
quaternionic form, 156
vector form, 123-125
Love, 13
Liiroth, 77

Macroscopic equations, 51
Magnetic current, 22, 30

induction, 23

Magnetization-electrons, 52, 56
Mass, and stress, 249

electromagnetic, 43, 215, 250

longitudinal and transversal, 196

rest-, 193

Mass-operator, 247
Matrices, 143-146, 160-162

alternating, 229

dual, 230
Matrix, embodying stress, energy,

etc., 238, 259, 278, 289
Matter and aether, 43
Maxwell, 7, 8, n, 24, 30, 32, 72, 88
Maxwell's equations/ 14, 20, 24

et seq., 261-262

Maxwellian stress, 48, 234, 286
Michelson, 41, 72, 77
Michelson and Morley, 41
Michelson-Morley experiment, 72-77
Microscopic (electronic) equations,

44, 205 et seq.

Minkowski, 6, 127, 129, 143, 195,
229, 260, 266, 268, 282, 284,
288

INDEX

293

Minkowski's electromagnetic equa-
tions, 267
representation of the Lorentz

transformation, 131-142
Moment, electric, 52
Moment of momentum, 5, 198

electromagnetic, 51
Momentum, electromagnetic, 50,

234, 281

of a particle, 195
Morley and Miller, 77
Mosengeil, 247
Motional electric force, 31

magnetic force, 31
Moving media, electromagnetic
equations for, 30 et seq., 260
et seq.

Moving mirror, 89
Multiplication of matrices, 160-161
of quaternions, 152-154

Natural periods, of vibration, 106

Newton, 249

Newtonian mechanics, 5, 15-17
transformation, 17, 115
system of reference, 5, 17

Newton's ' absolute time,' 7
third law, 45, 50

Non-Euclidean space, 176

Nordstrom, 241

Norm, of a quaternion, 154

Normal part, of a four-vector, 266,
288

Normal world-vectors, defined, 140

Nullifier, nullitat, 154

Ohm's law, 269

Optics of moving systems, 69-87
Orthogonality of world-vectors, 140,
146

Planck, 62, 245, 253, 256, 277
Poincare, 50, 87
Polar vectors, 146
Polarization-electrons, 52
Ponderable media, 260 et seq.
Ponderomotive forces, 44, 47-50,
218-223, 238, 240, 276 el seq.
Position-quaternion, 151
Postfactor, and prefactor, 222
Potential-quaternion, 217
Potentials, 80
Poynting, 47
Poynting and Gray, 249

and Thomson, 249
Poynting-vector, 47, 232, 286
Pressure, 48

isotropic, an invariant, 245
Principal axes, of mass-operator and

stress, 247

Principal diagonal, of a matrix, 161
Principle of areas, 198

of causality, 8

of conservation of energy, 283

of constant light-velocity, 99

of momentum, 283

of relativity, 99, no

of vis- viva, 195
Product of matrices, 160

of quaternions, 23, 152

scalar and vectoriai, 23
Propagation, 56-58
Proper time of a particle, 184, 195,

198

Pseudosphere, 178, 181
Ptolemy, 4

Pure electromagnetic waves, 209
Pythagoras, 4

Quaternions, elements of, 151-155
physical, 158
structure of, 159

Painleve, i, 14, 17

Parallelism, angle of, 180

Parameters, of a group of trans-
formations, 1 8, 125

Pender, 273

Period, relative and absolute, 69

Permeability (inductivity), 24

Permittivity, 24, 261, 267

Perpendicular world-vectors de-
fined, 140

Philolaus, 4

Physical bivectors, 208, 229
laws, in variance of, no
quaternions, 158, 199-204
time, 7-15

Radium, 256-257
Rapidity, defined, 179
Rayleigh, 83
Rays, luminous, 69
Reciprocal matrices, 144, 162
Reciprocal, of a quaternion, 154
Reciprocity of reference systems,

109
Reflection, from moving mirror,

89-91
Relative period, 69

stress, 243

Relativistic invariants, 112
Relativity of synchronism, 107
Relativity, principle of, 99

294

INDEX

Resistance, 269-270
Resultant force, 49-50

moment, 50

of rapidities, 179

of velocities, 164
Rest-acceleration, 187
Rest-mass, 193

anisotropy of, 249

electromagnetic, 216
Rest-system, 187
Reversibility, 253
Right-handed bivectors, 202
Ritz, 73
Robb, 179
Romer, 35
Rontgen, 32, 275
Rotation, 5

obtained by a pair of quaternions,
209

four-dimensional, 127, 149
Rowland, 153

Scalar part of a quaternion, 151

potential, 80, 217

product, 23

product, four-dimensional, 148
Schiaparelli, 38
Schuster, 43
Simultaneity, Einstein's definition

of, 93-98
Singular quaternions, 154, 159

vectors, 135, 141

Six-parametric Lorentz transfor-
mation, 167, 170
Six-vectors, 147, 229

special, 201

Size, of a four-vector, 148
Solar system, motion of, 17, 88
Sommerfeld, 140, 146, 175, 176,

245
Space-like quaternions, 159

vectors, 135, 141
Space-time filament, 130

vectors, of the first kind, 140

of the second kind, 229
State, defined, 9
States, lines of, 19
Stokes' aether, 42, 62, 63
Stress, electromagnetic, 48, 234,

281

Stress-energy matrix, 278, 289
Stress, relative and absolute, 243-

244

Stretcher (operator), 125
Structure, of a quaternion, 159
Sub-group, Lorentz', 127-129
Sum of matrices, 160

of quaternions, 152

Sun, radiation of, 256
Supplement, of a special bivector,

201

Sutherland, 77
Synchronism, relative, 107
Synchronous clocks, 95
System of reference, 2-6, 17, 1 8
Systems, complete, 9, 18-20
incomplete, 10-14

Tait, 209

Temperature, transformation of,

253

Tension, 48

Tensor, of a quaternion, 153

Terrestrial optics, 71

Thales, 4

Theorem of corresponding states,

67-71

Thermodynamical properties, 253
Thomson, 34, 250

Three-parametric Lorentz trans-
formations, 170
Time, effective, 85

electromagnetic, 14

kinetic, 13

local, 66, 84

Newton's, 7

proper, 184, 195, 198
Time-keeper, choice of, 6-15
Time-like quaternions, 159

vectors, 134, 141
Times compared, 101-107
Timerding, 130
Total current, 265
Transformation, Lorentz-, Cartesian
form, 86, no

geometric representation, 131-
142

matrix form, 144

quaternionic form, 156

vector form, 123-125
Transformation, Newtonian, 17

of stress, etc., 236 et seq.
Transposed matrix, 144, 160
Transversal discontinuity, 57

mass, 196, 255

electromagnetic, 215
Triangle, Lobatchewskyan, 177,

178, 181
Trouton and Noble, 250

Undisturbed systems, 9, 18-20
Uniform motion, 16-18
Unit-matrix, 162

quaternion, 154

rapidity, 180

INDEX

295

Universal time (Lorentz), 64

Vacuum-equations, 26, 269

Varicak, 38, 176

Vectors, axial and polar, 146

localized, pseudospherical, 179

singular, 135, 141

space-like, 135

time-like, 134
Vector part of a quaternion, 151

potential, 80, 217

product, 23

Velocities, compounded, 116, 126
Velocity of propagation, 28, 56

of light, 23, 33, 73, 88, 171, 172,
180, 181

constant, principle of, 99
Velocity-quaternion, 183
Versor, of a quaternion, 154
Voigt, 85

Water- telescope experiment, 38-39,

60-62

Wave, pure, 209
Wave-normals, 69
Waves of discontinuity, 28, 29,

56-58, 120
Wells, 134
Whittaker, 36
Wilson, H. A., 32, 274
Wilson, E. B., and Lewis, 150
Woods, 177

World, four-dimensional, 4, 129
World-line, 130, 185

hyperbolic curvature of, 191
World-point, 129

tensor, 239

tube, 130

vectors, denned, 140
Worthington, 5

Young, 35, 36

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